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Hypothesis testing basics: one tail or two, left tailed test or right tailed test example, how to run a right tailed test.
In a hypothesis test , you have to decide if a claim is true or not. Before you can figure out if you have a left tailed test or right tailed test, you have to make sure you have a single tail to begin with. A tail in hypothesis testing refers to the tail at either end of a distribution curve.
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If you can sketch a graph, you can figure out which tail is in your test. Back to Top
Example question: You are testing the hypothesis that the drop out rate is more than 75% (>75%). Is this a left-tailed test or a right-tailed test?
Step 1: Write your null hypothesis statement and your alternate hypothesis statement. This step is key to drawing the right graph, so if you aren’t sure about writing a hypothesis statement, see: How to State the Null Hypothesis.
Step 2: Draw a normal distribution curve.
Step 3: Shade in the related area under the normal distribution curve . The area under a curve represents 100%, so shade the area accordingly. The number line goes from left to right, so the first 25% is on the left and the 75% mark would be at the left tail.
The yellow area in this picture illustrates the area greater than 75%. Left Tailed Test or Right Tailed Test? From this diagram you can clearly see that it is a right-tailed test, because the shaded area is on the right .
That’s it!
Hypothesis tests can be three different types:
The right tailed test and the left tailed test are examples of one-tailed tests . They are called “one tailed” tests because the rejection region (the area where you would reject the null hypothesis ) is only in one tail. The two tailed test is called a two tailed test because the rejection region can be in either tail.
Here’s what the right tailed test looks like on a graph:
A right tailed test (sometimes called an upper test) is where your hypothesis statement contains a greater than (>) symbol. In other words, the inequality points to the right. For example, you might be comparing the life of batteries before and after a manufacturing change. If you want to know if the battery life is greater than the original (let’s say 90 hours), your hypothesis statements might be: Null hypothesis : No change or less than (H 0 ≤ 90). Alternate hypothesis : Battery life has increased (H 1 ) > 90.
The important factor here is that the alternate hypothesis (H 1 ) determines if you have a right tailed test, not the null hypothesis .
A high-end computer manufacturer sets the retail cost of their computers based in the manufacturing cost, which is $1800. However, the company thinks there are hidden costs and that the average cost to manufacture the computers is actually much more. The company randomly selects 40 computers from its facilities and finds that the mean cost to produce a computer is $1950 with a standard deviation of $500. Run a hypothesis test to see if this thought is true.
Step 1: Write your hypothesis statement (see: How to state the null hypothesis ). H 0 : μ ≤ 1800 H 1 : μ > 1800
Step 3: Choose an alpha level . No alpha is mentioned in the question, so use the standard (0.05). 1 – 0.05 = .95 Look up that value (.95) in the middle of the z-table. The area corresponds to a z-value of 1.645. That means you would reject the null hypothesis if your test statistic is greater than 1.645.*
1.897 is greater than 1.645, so you can reject the null hypothesis .
* Not sure how I got 1.645? The left hand half of the curve is 50%, so you look up 45% in the “right of the mean” table on this site (50% + 45% = 95%).
This z-table shows the area to the right of the mean , so you’re actually looking up .45, not .95. That’s because half of the area (.5) is not actually showing on the table.
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Dodge, Y. (2008). The Concise Encyclopedia of Statistics . Springer. Gonick, L. (1993). The Cartoon Guide to Statistics . HarperPerennial. “Klein, G. (2013). The Cartoon Introduction to Statistics. Hill & Wamg. Kotz, S.; et al., eds. (2006), Encyclopedia of Statistical Sciences , Wiley. Wheelan, C. (2014). Naked Statistics . W. W. Norton & Company
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The p-value in statistics quantifies the evidence against a null hypothesis. A low p-value suggests data is inconsistent with the null, potentially favoring an alternative hypothesis. Common significance thresholds are 0.05 or 0.01.
When you perform a statistical test, a p-value helps you determine the significance of your results in relation to the null hypothesis.
The null hypothesis (H0) states no relationship exists between the two variables being studied (one variable does not affect the other). It states the results are due to chance and are not significant in supporting the idea being investigated. Thus, the null hypothesis assumes that whatever you try to prove did not happen.
The alternative hypothesis (Ha or H1) is the one you would believe if the null hypothesis is concluded to be untrue.
The alternative hypothesis states that the independent variable affected the dependent variable, and the results are significant in supporting the theory being investigated (i.e., the results are not due to random chance).
A p-value, or probability value, is a number describing how likely it is that your data would have occurred by random chance (i.e., that the null hypothesis is true).
The level of statistical significance is often expressed as a p-value between 0 and 1.
The smaller the p -value, the less likely the results occurred by random chance, and the stronger the evidence that you should reject the null hypothesis.
Remember, a p-value doesn’t tell you if the null hypothesis is true or false. It just tells you how likely you’d see the data you observed (or more extreme data) if the null hypothesis was true. It’s a piece of evidence, not a definitive proof.
Suppose you’re conducting a study to determine whether a new drug has an effect on pain relief compared to a placebo. If the new drug has no impact, your test statistic will be close to the one predicted by the null hypothesis (no difference between the drug and placebo groups), and the resulting p-value will be close to 1. It may not be precisely 1 because real-world variations may exist. Conversely, if the new drug indeed reduces pain significantly, your test statistic will diverge further from what’s expected under the null hypothesis, and the p-value will decrease. The p-value will never reach zero because there’s always a slim possibility, though highly improbable, that the observed results occurred by random chance.
The significance level (alpha) is a set probability threshold (often 0.05), while the p-value is the probability you calculate based on your study or analysis.
A p-value less than or equal to a predetermined significance level (often 0.05 or 0.01) indicates a statistically significant result, meaning the observed data provide strong evidence against the null hypothesis.
This suggests the effect under study likely represents a real relationship rather than just random chance.
For instance, if you set α = 0.05, you would reject the null hypothesis if your p -value ≤ 0.05.
It indicates strong evidence against the null hypothesis, as there is less than a 5% probability the null is correct (and the results are random).
Therefore, we reject the null hypothesis and accept the alternative hypothesis.
Upon analyzing the pain relief effects of the new drug compared to the placebo, the computed p-value is less than 0.01, which falls well below the predetermined alpha value of 0.05. Consequently, you conclude that there is a statistically significant difference in pain relief between the new drug and the placebo.
A p-value of 0.001 is highly statistically significant beyond the commonly used 0.05 threshold. It indicates strong evidence of a real effect or difference, rather than just random variation.
Specifically, a p-value of 0.001 means there is only a 0.1% chance of obtaining a result at least as extreme as the one observed, assuming the null hypothesis is correct.
Such a small p-value provides strong evidence against the null hypothesis, leading to rejecting the null in favor of the alternative hypothesis.
This means we retain the null hypothesis and reject the alternative hypothesis. You should note that you cannot accept the null hypothesis; we can only reject it or fail to reject it.
Note : when the p-value is above your threshold of significance, it does not mean that there is a 95% probability that the alternative hypothesis is true.
Most statistical software packages like R, SPSS, and others automatically calculate your p-value. This is the easiest and most common way.
Online resources and tables are available to estimate the p-value based on your test statistic and degrees of freedom.
These tables help you understand how often you would expect to see your test statistic under the null hypothesis.
Understanding the Statistical Test:
Different statistical tests are designed to answer specific research questions or hypotheses. Each test has its own underlying assumptions and characteristics.
For example, you might use a t-test to compare means, a chi-squared test for categorical data, or a correlation test to measure the strength of a relationship between variables.
Be aware that the number of independent variables you include in your analysis can influence the magnitude of the test statistic needed to produce the same p-value.
This factor is particularly important to consider when comparing results across different analyses.
If you’re comparing the effectiveness of just two different drugs in pain relief, a two-sample t-test is a suitable choice for comparing these two groups. However, when you’re examining the impact of three or more drugs, it’s more appropriate to employ an Analysis of Variance ( ANOVA) . Utilizing multiple pairwise comparisons in such cases can lead to artificially low p-values and an overestimation of the significance of differences between the drug groups.
A statistically significant result cannot prove that a research hypothesis is correct (which implies 100% certainty).
Instead, we may state our results “provide support for” or “give evidence for” our research hypothesis (as there is still a slight probability that the results occurred by chance and the null hypothesis was correct – e.g., less than 5%).
In our comparison of the pain relief effects of the new drug and the placebo, we observed that participants in the drug group experienced a significant reduction in pain ( M = 3.5; SD = 0.8) compared to those in the placebo group ( M = 5.2; SD = 0.7), resulting in an average difference of 1.7 points on the pain scale (t(98) = -9.36; p < 0.001).
The 6th edition of the APA style manual (American Psychological Association, 2010) states the following on the topic of reporting p-values:
“When reporting p values, report exact p values (e.g., p = .031) to two or three decimal places. However, report p values less than .001 as p < .001.
The tradition of reporting p values in the form p < .10, p < .05, p < .01, and so forth, was appropriate in a time when only limited tables of critical values were available.” (p. 114)
A lower p-value is sometimes interpreted as meaning there is a stronger relationship between two variables.
However, statistical significance means that it is unlikely that the null hypothesis is true (less than 5%).
To understand the strength of the difference between the two groups (control vs. experimental) a researcher needs to calculate the effect size .
In statistical hypothesis testing, you reject the null hypothesis when the p-value is less than or equal to the significance level (α) you set before conducting your test. The significance level is the probability of rejecting the null hypothesis when it is true. Commonly used significance levels are 0.01, 0.05, and 0.10.
Remember, rejecting the null hypothesis doesn’t prove the alternative hypothesis; it just suggests that the alternative hypothesis may be plausible given the observed data.
The p -value is conditional upon the null hypothesis being true but is unrelated to the truth or falsity of the alternative hypothesis.
If your p-value is less than or equal to 0.05 (the significance level), you would conclude that your result is statistically significant. This means the evidence is strong enough to reject the null hypothesis in favor of the alternative hypothesis.
No, not all p-values below 0.05 are considered statistically significant. The threshold of 0.05 is commonly used, but it’s just a convention. Statistical significance depends on factors like the study design, sample size, and the magnitude of the observed effect.
A p-value below 0.05 means there is evidence against the null hypothesis, suggesting a real effect. However, it’s essential to consider the context and other factors when interpreting results.
Researchers also look at effect size and confidence intervals to determine the practical significance and reliability of findings.
Sample size can impact the interpretation of p-values. A larger sample size provides more reliable and precise estimates of the population, leading to narrower confidence intervals.
With a larger sample, even small differences between groups or effects can become statistically significant, yielding lower p-values. In contrast, smaller sample sizes may not have enough statistical power to detect smaller effects, resulting in higher p-values.
Therefore, a larger sample size increases the chances of finding statistically significant results when there is a genuine effect, making the findings more trustworthy and robust.
No, a non-significant p-value does not necessarily indicate that there is no effect or difference in the data. It means that the observed data do not provide strong enough evidence to reject the null hypothesis.
There could still be a real effect or difference, but it might be smaller or more variable than the study was able to detect.
Other factors like sample size, study design, and measurement precision can influence the p-value. It’s important to consider the entire body of evidence and not rely solely on p-values when interpreting research findings.
While a p-value can be extremely small, it cannot technically be absolute zero. When a p-value is reported as p = 0.000, the actual p-value is too small for the software to display. This is often interpreted as strong evidence against the null hypothesis. For p values less than 0.001, report as p < .001
Bland, J. M., & Altman, D. G. (1994). One and two sided tests of significance: Authors’ reply. BMJ: British Medical Journal , 309 (6958), 874.
Goodman, S. N., & Royall, R. (1988). Evidence and scientific research. American Journal of Public Health , 78 (12), 1568-1574.
Goodman, S. (2008, July). A dirty dozen: twelve p-value misconceptions . In Seminars in hematology (Vol. 45, No. 3, pp. 135-140). WB Saunders.
Lang, J. M., Rothman, K. J., & Cann, C. I. (1998). That confounded P-value. Epidemiology (Cambridge, Mass.) , 9 (1), 7-8.
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In this post, you’ll learn how to perform t-tests in Python using the popular SciPy library . T-tests are used to test for statistical significance and can be hugely advantageous when working with smaller sample sizes.
By the end of this tutorial, you’ll have learned the following:
Table of Contents
The t-test, or often referred to as the student’s t-test , dates back to the early 20th century. An Irish statistician working for Guinness Brewery, William Sealy Gosset, introduced the concept. Because the brewery was working with small sample sizes and was under strict orders of confidentiality, Gosset published his findings under the pseudonym “Student”. His seminal work, “The Probable Error of a Mean,” laid the groundwork for what we now know as Student’s t-test.
This leads us to one of the primary benefits of the t-test: the t-test is able to make reliable inferences about a population using a small sample size . Let’s explore how this works by discussing the theory behind the t-test in the following section.
Statistical tests are used to make assumptions about some population parameters. For example, it lets us test whether or not the average test score for any given group of students is 70%. The T-Test works in two different ways:
Let’s explore these in a little more depth.
The one-sample t-test is used to test the null hypothesis that the population mean inferred from a sample is equal to some given value. It can be described as below:
There are actually three different alternative hypotheses:
We can use the following formula to calculate our test statistic:
We then need to calculate the p-value using degrees of freedom equal to n – 1. If the p-value is less than your chosen significance level, we can reject the null hypothesis and say that the means differ.
The two-sample t-test is used to test whether two population means are equal (or if they differ in a significant way). In this case, the null hypothesis assumes that the two population means are equal.
When we sample two different groups, we are almost guaranteed that their sample means will differ. But the t-test allows us to test whether or not this difference is different in a statistically significant way.
Similar to the one-sample t-test, there are three different alternative hypotheses:
The formula for the two-sample t-test can be written as:
We then need to calculate the p-value using degrees of freedom equal to (n 1 +n 2 -1). If the p-value is less than your chosen significance level, we can reject the null hypothesis and say that the means differ.
Both types of t-tests follow a key set of assumptions, including:
It’s easy to test for these assumptions using Python (and I have included links to tutorials covering how to do this). Let’s take a look at example walkthroughs of how to conduct both of these tests in Python.
In this section, you’ll learn how to conduct a one-sample t-test in Python. Suppose you are a teacher and have just given a test. You know that the population mean for this test is 85% and you want to see whether the score of the class is significantly different from this population mean.
Let’s start by importing our required function, ttest_1samp() from SciPy and defining our data:
In the code block above, we first imported our required library. We then defined our sample as a list of values and defined our population mean as its own variable.
We can now pass these values into the function, as shown below:
The function returns a test statistic and the corresponding p-value. We can print these values out using f-strings to simplify the labeling , as shown above.
Finally, we can write a simple if-else statement to evaluate whether or not our sample mean is significantly different from the population mean:
We can see that by running this if-else statement, that our test indicates that there is no significant difference in the exam scores.
In order to calculate the different one-sample t-test alternative hypotheses, we can use the alternative= parameter:
Now that you have a strong understanding of how to perform a one-sample t-test, let’s dive into the exciting world of two-sample t-tests!
A two-sample t-test is used to test whether the means of two samples are equal. The test requires that both samples be normally distributed, have similar variances, and be independent of one another.
Imagine that we want to compare the test scores of two different classes. This is the perfect example of when to use a t-test. Let’s begin by running a two-tailed test, which only evaluates whether or not the two means are equal. It begins with the null hypothesis, which states that the two means are equal.
Let’s take a look at how we can run a two-tailed t-test in Python:
We can see that the ttest_ind() function returns both a test statistic and a p-value. We can run a simple if-else statement to check whether or not we can reject or fail to reject the null hypothesis:
We can see that there is a significant difference between the two sets of scores. However, the two-tailed test doesn’t tell us in which direction.
In order to do this, we need to use a right- or left-tailed two-sample t-test. To do this in SciPy, we use the alternative= parameter. By default, this is set to 'two-sided' . However, we can modify this to either 'less' or 'greater' , if we want to evaluate whether or not the mean for one sample is less than or greater than another.
Let’s see how we can check if the mean of class 2 is significantly higher than that of class 1:
Because our p-value is less than our defined value of 0.05, we can say that the mean of class 2 is higher with statistical significance.
In conclusion, this comprehensive guide has equipped you with the knowledge and practical skills to perform t-tests in Python using the SciPy library. T-tests are invaluable tools for assessing statistical significance, particularly when working with smaller sample sizes.
Throughout this tutorial, you’ve gained insights into:
Remember that t-tests come with certain assumptions, and it’s crucial to validate them before applying these tests to your data. Python provides tools to check these assumptions, ensuring the robustness and reliability of your statistical analyses.
To learn more about these functions, check out the official documentation for the one-sample t-test and for the two-sample t-test in SciPy.
Nik is the author of datagy.io and has over a decade of experience working with data analytics, data science, and Python. He specializes in teaching developers how to use Python for data science using hands-on tutorials. View Author posts
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A one-tailed test is a statistical test in which the critical area of a distribution is one-sided so that it is either greater than or less than a certain value, but not both. If the sample being tested falls into the one-sided critical area, the alternative hypothesis will be accepted instead of the null hypothesis.
Financial analysts use the one-tailed test to test an investment or portfolio hypothesis.
A basic concept in inferential statistics is hypothesis testing . Hypothesis testing is run to determine whether a claim is true or not, given a population parameter. A test that is conducted to show whether the mean of the sample is significantly greater than and significantly less than the mean of a population is considered a two-tailed test . When the testing is set up to show that the sample mean would be higher or lower than the population mean, it is referred to as a one-tailed test. The one-tailed test gets its name from testing the area under one of the tails (sides) of a normal distribution , although the test can be used in other non-normal distributions.
Before the one-tailed test can be performed, null and alternative hypotheses must be established. A null hypothesis is a claim that the researcher hopes to reject. An alternative hypothesis is the claim supported by rejecting the null hypothesis.
A one-tailed test is also known as a directional hypothesis or directional test.
Let's say an analyst wants to prove that a portfolio manager outperformed the S&P 500 index in a given year by 16.91%. They may set up the null (H 0 ) and alternative (H a ) hypotheses as:
H 0 : μ ≤ 16.91
H a : μ > 16.91
The null hypothesis is the measurement that the analyst hopes to reject. The alternative hypothesis is the claim made by the analyst that the portfolio manager performed better than the S&P 500. If the outcome of the one-tailed test results in rejecting the null, the alternative hypothesis will be supported. On the other hand, if the outcome of the test fails to reject the null, the analyst may carry out further analysis and investigation into the portfolio manager’s performance.
The region of rejection is on only one side of the sampling distribution in a one-tailed test. To determine how the portfolio’s return on investment compares to the market index, the analyst must run an upper-tailed significance test in which extreme values fall in the upper tail (right side) of the normal distribution curve. The one-tailed test conducted in the upper or right tail area of the curve will show the analyst how much higher the portfolio return is than the index return and whether the difference is significant.
The most common significance levels (p-values) used in a one-tailed test.
To determine how significant the difference in returns is, a significance level must be specified. The significance level is almost always represented by the letter p, which stands for probability. The level of significance is the probability of incorrectly concluding that the null hypothesis is false. The significance value used in a one-tailed test is either 1%, 5%, or 10%, although any other probability measurement can be used at the discretion of the analyst or statistician. The probability value is calculated with the assumption that the null hypothesis is true. The lower the p-value , the stronger the evidence that the null hypothesis is false.
If the resulting p-value is less than 5%, the difference between both observations is statistically significant, and the null hypothesis is rejected. Following our example above, if the p-value = 0.03, or 3%, then the analyst can be 97% confident that the portfolio returns did not equal or fall below the return of the market for the year. They will, therefore, reject H 0 and support the claim that the portfolio manager outperformed the index. The probability calculated in only one tail of a distribution is half the probability of a two-tailed distribution if similar measurements were tested using both hypothesis testing tools.
When using a one-tailed test, the analyst is testing for the possibility of the relationship in one direction of interest and completely disregarding the possibility of a relationship in another direction. Using our example above, the analyst is interested in whether a portfolio’s return is greater than the market’s. In this case, they do not need to statistically account for a situation in which the portfolio manager underperformed the S&P 500 index. For this reason, a one-tailed test is only appropriate when it is not important to test the outcome at the other end of a distribution.
A one-tailed test looks for an increase or decrease in a parameter. A two-tailed test looks for change, which could be a decrease or an increase.
A one-tailed T-test checks for the possibility of a one-direction relationship but does not consider a directional relationship in another direction.
You would use a two-tailed test when you want to test your hypothesis in both directions.
University of Southern California. " FAQ: What Are the Differences Between One-Tailed and Two-Tailed Tests? "
I'll answer that question, explain a statistical test you might not have heard of, and introduce you to my new obsession: bridge..
After leaving my last job, I started playing a lot of contract bridge (or “contact bridge” as it’s called in my family, as games tend to get heated). I recently participated in a regional bridge tournament. I was inspired to write this post about p -values when I found myself in what I thought was a very unlikely situation during a bridge session.
I could talk bridge all day, but I know it’s not for everyone. Given that, I’ll try to minimize the background for this article. Bridge is a trick-taking card game played with a standard deck of 52 cards. There are four players seated around the table: North, East, South, and West. North and South are partners, and East and West are partners. Each hand is preceded by an auction in which the players bid on 1) how many tricks they think they can win, and 2) which suit is the trump suit. For example, if North-South wins the auction with a bid of 4 hearts, then they claim that they will take 6 + 4 = 10 tricks with hearts as trump. (You add 6 because there are 52/4 = 13 total tricks, so it doesn’t make sense to proclaim that you’ll take fewer than half.)
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The weird thing about bridge (well, one weird thing) is that only three of the four players play each hand. One person from the team that wins the auction plays the hand. That person is called the “declarer.” 1 The declarer’s partner is called the “dummy,” and the dummy’s hand is placed face up after the opening lead by the defense. The declarer then plays both hands, concealing their own. All eyes are on you as the declarer, so that is the most exciting and stressful position to be in. For concreteness, suppose that North is the declarer. This means that South, North’s partner, is the dummy, while East and West play defense. Defense always plays the first card, and they lead into the dummy. Therefore, East would lead in this case. As soon as East plays their first card, South reveals their hand, and then play continues clockwise around the compass (with North playing both North’s and South’s hands). After all 13 tricks are played, you count how many tricks the declarer took and award points according to how they fared against their bid.
So what does this have to do with p -values? As I said above, bridge is most exciting when you are the declarer. In one of my games during the recent tournament, my team played 24 hands, and I noticed that I was starting to get bored. It turns out that I was the declarer on only two of the 24 hands! Two out of 24 is 8.3%, which seemed like a very small percentage to me. Since North, East, South, and West are just names, you expect each player should declare 25% of the time in the long run. This made me wonder if chance was to blame, or rather if something about the bidding habits of the people at the table skewed the results.
Statistics gives us a way of understanding how unlikely experiences like my bridge game are. Let’s assume that North really should play 25% of hands. In that case, the proportion of hands that North plays out of N = 24 random deals is approximately normally distributed. This is a consequence of the central limit theorem . The mean of that normal distribution is π = .25, and the standard deviation is
Knowing this, we can compute the z -score of the sample proportion p = 2/24 = 8.3%:
This means that the observed proportion of 2/24 hands is almost two standard deviations below the mean of 6/24 hands. The next step is to convert this z -score into a p -value . The p -value is the area under the standard normal (i.e., bell-shaped) curve to the left of z = -1.889. You can compute it Excel with the function NORM.S.DIST(-1.889, 1). The 1 means that you want the cumulative probability (i.e., area) and not the height of the curve at that point. Entering that function returns a p-value of about .0297 (cf., picture below).
The next question is, how do we interpret the p -value? Before getting to that, it’s best to explain the context in which p -values most often arise. Hypothesis testing is the branch of statistical inference wherein you collect samples from a population to test an assumption about the population. In this case, the null hypothesis is that the true proportion of hands with North as declarer is 25%. The alternative hypothesis—based on our observation—is that North actually declares less than 25% of the time. A good real-life example is a clinical trial for a drug. In that case, you have a treatment and control group. The null hypothesis is that there is no difference in outcomes (measured however you want) between those two groups. The alternative hypothesis is that the treatment group fares better than the control group. In either case, the p -value is interpreted as follows:
Assuming that the null hypothesis is true, there is a p% chance that a random sample of this size would be as extreme as this sample.
Applied to the bridge case: Assuming that North declares on 25% of hands, there is a 2.97% chance that North would declare two or fewer times in a game with 24 hands. The 24 hands part is critical; the standard deviation of the sampling distribution depends on N , the number of hands. The more hands you play, the more unlikely it would be for North to declare in fewer than 8.3% of hands. In a hypothesis test, you typically have some cutoff—called α—that you compare to the p- value. For example, if α = .01, then you would say that a p -value less than 1% is just too unlikely to believe. The only possible conclusion is that your assumption—the null hypothesis—must be false. Depending on the context, α can be .001, .01, .05, or sometimes .1.
There are a few common misconceptions about the p -value. The first is that it depends on α. As we saw, you can define the p -value without any reference to α, which is simply your line in the sand for what constitutes “beyond a reasonable doubt.” Another common misconception is that the p -value is the probability that the null hypothesis is true. That’s not right. All the p -value tells you is the probability of observing your sample, if the null hypothesis is true . Finally, some people claim that the smaller the p -value, the stronger the effect (in a treatment/control scenario). This is also not true. The p -value just measures how unlikely it is to observe a sample. Suppose you knew that a drug reduced someone’s cholesterol by an average of 15 mg/dL. If you repeatedly ran controlled experiments, you could make the p -value as small as you want by increasing the sizes of the treatment and control groups (thereby reducing variability). However, the magnitude of the effect will always be 15 mg/dL, on average, regardless of the p -value.
With a p -value of under 3%, it’s tempting to argue that declaring on two out of 24 hands is too rare to attribute to chance. 2 However, I didn’t share all the details of my game. Of the 24 hands, I (North) declared twice, East five times, South eight times, and West nine times. As discussed, each player expects to be declarer on approximately 25% of hands. There’s another statistical test— χ -square goodness of fit —which can be used to analyze the distribution of categorical variables. This test works by comparing the observed counts of each possible value of a variable to the expected counts (6 each for North, South, East, and West in our case). To avoid positive and negative differences offsetting, each error is squared. The squared differences are then divided by the expected count and added together. (This is not unlike linear regression, which I discussed in this previous post .) The result is a single number called the χ -square statistic. This number is always positive, and it captures how far away the distribution of a variable is from what is expected. Looking at the table below, we see that χ -square = 5.0 in this case.
To compute a p -value, we need to know what type of distribution χ -square follows. The central limit theorem says that, for each player, the square root of Column F in the above picture converges to the standard normal normal as N increases. 3 This makes χ -square the sum of squares of normally distributed variables. Shockingly, such sums follow a neat distribution called the χ -square distribution (hence the name of the statistic/test). Once you know that, then the calculation of the p -value and its interpretation are exactly the same. To calculate the p-value, you can type
in Excel. 5 is the value of the χ -square statistic. 3 is the degrees of freedom (number of players - 1) and 1 means cumulative, as with NORM.S.DIST. We have to subtract the value from 1 because now the extreme side is the right tail, not the left tail.
Notice that the p -value from the χ -square test is much larger: 17.2%. This number is large enough that we would not be able to reject the null hypothesis, which is that each player plays 25% of the hands. So, on the one hand, the first test says that it is very unlikely (about 3 out of 100) that North would play zero, one, or two out of 24 hands. On the other hand, the χ -square test says that an arrangement as uneven as (N, E, S, W) = (2, 5, 8, 9) is not all that rare: it would happen more than one in six times that you play 24 hands.
This begs the question, which test is right? The short answer is that they’re both right. They’re just answering different questions. In the first case, I zoomed in on the experience of North. From North’s perspective, it’s true that it’s very rare to declare two or fewer times out of 24 hands. However, declaring is a zero-sum game (well, I guess a 24-sum game): if any of the other seats declare more times than expected, that means someone has to declare fewer times than expected. The χ -square test considers all four hands together. When you do that, the arrangement (2, 8, 9, 5) is seen to be not that pathological. In fact, I played six times during the tournament, so the chances are good that I would see some declarer distribution with p = .172 at least once. Reflecting on the tournament now, I can’t remember the number of times I declared in any of the other sessions. You could argue that I engaged in “ p hacking” by focusing on the one game in which I declared only twice. p hacking is a dishonest (yet common) approach to statistical analysis in which you take multiple samples until you get a p -value that supports the conclusion you want to draw.
To close, I want to clarify one thing about the χ -square statistic. It reduces the deviation from the expected counts for all players into a single number. For example, (3, 3, 9, 9) has a higher χ -square value of 6.0 (thus, a lower p -value), even though no one declared two or fewer times. If you wanted to get the exact probability of someone declaring two or fewer times, you would apply the multinomial distribution , which models the different ways you can add up four numbers to get 24. This is tricky, though, because it’s difficult to systematically list the arrangements (N, E, S, W) that have a minimum value of 0, 1, or 2 in one of the hands. 4 Instead, I simulated 500 sessions of 24 hands and noted 1) the percentage of sessions that each player played two or fewer hands, and 2) the percentage of sessions that each of 0, 1, 2, 3, 4, 5, and 6 was the minimum number of hands declared by any player. (The minimum can’t be greater than 6 because the average is 6.) As you see below, the minimum number of hands played was two or fewer in 16.8% of the 500 sessions, which is slightly less than the χ -square test. I re-ran this experiment another ~30 times, and the average was closer to 15.6%.
That’s all for today. Thank you for reading and indulging my bridge obsession. Please subscribe and share if you enjoyed this. Most importantly, please let me know if you’re looking for a bridge partner.
The declarer is the first person on the declaring team to mention the ultimate trump suit in the auction. For example, if North opens the bidding with 1 heart, they would play the hand if North-South ends up winning the auction with hearts as the trump suit. This is true even if South makes the final bid of 4 hearts.
Some possible explanations: the hands aren’t random, our bidding or our opponents bidding is unusual, etc.
Briefly, if you convert the counts to frequencies, you get a binomial distribution, and then it’s easier to see how to apply CLT. If you want more rigor, don’t read blogs. (Just kidding… here’s a proof .)
The “ stars and bars ” method tells you that there are 2,925 (= 27 choose 3) ways that N-E-S-W can share declarer in 24 hands. There are probably several hundred that have at least one 0, 1, or 2 in one of the positions. I was not up to the task of enumerating them.
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Two-tailed hypothesis tests are also known as nondirectional and two-sided tests because you can test for effects in both directions. When you perform a two-tailed test, you split the significance level percentage between both tails of the distribution. In the example below, I use an alpha of 5% and the distribution has two shaded regions of 2. ...
To test this, she can perform a one-tailed hypothesis test with the following null and alternative hypotheses: H 0 (Null Hypothesis): μ = 10 inches; H A (Alternative Hypothesis): μ ≠ 10 inches; This is an example of a two-tailed hypothesis test because the alternative hypothesis contains the not equal "≠" sign. The botanist believes ...
Two-Tailed Test: A two-tailed test is a statistical test in which the critical area of a distribution is two-sided and tests whether a sample is greater than or less than a certain range of values ...
A two tailed test tells you that you're finding the area in the middle of a distribution. In other words, your rejection region (the place where you would reject the null hypothesis) is in both tails. For example, let's say you were running a z test with an alpha level of 5% (0.05). In a one tailed test, the entire 5% would be in a single tail.
In coin flipping, the null hypothesis is a sequence of Bernoulli trials with probability 0.5, yielding a random variable X which is 1 for heads and 0 for tails, and a common test statistic is the sample mean (of the number of heads) ¯. If testing for whether the coin is biased towards heads, a one-tailed test would be used - only large numbers of heads would be significant.
A two-sample t-test always uses the following null hypothesis: H 0: μ 1 = μ 2 (the two population means are equal) The alternative hypothesis can be either two-tailed, left-tailed, or right-tailed: H 1 (two-tailed): μ 1 ≠ μ 2 (the two population means are not equal) H 1 (left-tailed): μ 1 < μ 2 (population 1 mean is less than population ...
A two-tailed test will test both if the mean is significantly greater than x and if the mean significantly less than x. ... So, depending on the direction of the one-tailed hypothesis, its p-value is either .5*(two-tailed p-value) or 1-.5*(two-tailed p-value) if the test statistic symmetrically distributed about zero.
A one tailed test does not leave more room to conclude that the alternative hypothesis is true. The benefit (increased certainty) of a one tailed test doesn't come free, as the analyst must know "something more", which is the direction of the effect, compared to a two tailed test. Show more...
The alternate hypothesis for a two-sided t-test would simply state that the mean blood pressure for the medication group is different than the placebo group, but it wouldn't specify if medication would raise or lower the mean blood pressure. Typically, researchers choose to use two-sided t-tests, since they usually don't know how a ...
So if the alternate hypothesis is written with a ≠ sign that means that we are going to perform a 2-tailed test because chances are it could be more than 100 or less than 100 which makes it 2-tailed. So, after stating the Null and Alternative hypothesis, it's time to move to step-2 which is: 2. Choose the level of Significance(α)
The two red tails are the alpha level, divided by two (i.e. α/2). Alpha levels (sometimes just called "significance levels") are used in hypothesis tests; it is the probability of making the wrong decision when the null hypothesis is true. A one-tailed test has the entire 5% of the alpha level in one tail (in either the left, or the right tail).
The procedure for hypothesis testing is based on the ideas described above. Specifically, we set up competing hypotheses, select a random sample from the population of interest and compute summary statistics. ... In a two-tailed test the decision rule has investigators reject H 0 if the test statistic is extreme, either larger than an upper ...
This test is called a directional or one‐tailed test because the region of rejection is entirely within one tail of the distribution. Some hypotheses predict only that one value will be different from another, without additionally predicting which will be higher. The test of such a hypothesis is nondirectional or two‐tailed because an ...
Two-Tailed. In our example concerning the mean grade point average, suppose again that our random sample of n = 15 students majoring in mathematics yields a test statistic t* instead of equaling -2.5.The P-value for conducting the two-tailed test H 0: μ = 3 versus H A: μ ≠ 3 is the probability that we would observe a test statistic less than -2.5 or greater than 2.5 if the population mean ...
There are three different types of hypothesis tests: Two-tailed test: The alternative hypothesis contains the "≠" sign. Left-tailed test: The alternative hypothesis contains the "<" sign. Right-tailed test: The alternative hypothesis contains the ">" sign. Notice that we only have to look at the sign in the alternative hypothesis ...
A one-tailed test looks for an increase or decrease in the parameter whereas a two-tailed test looks for any change in the parameter (which can be any change- increase or decrease). We can perform the test at any level (usually 1%, 5% or 10%). For example, performing the test at a 5% level means that there is a 5% chance of wrongly rejecting H 0.
Step 2: Collect data. For a statistical test to be valid, it is important to perform sampling and collect data in a way that is designed to test your hypothesis. If your data are not representative, then you cannot make statistical inferences about the population you are interested in. Hypothesis testing example.
The one-tailed hypothesis is rejected only if the sample proportion is much greater than \(0.5\). The alternative hypothesis in the two-tailed test is \(\pi \neq 0.5\). In the one-tailed test it is \(\pi > 0.5\). You should always decide whether you are going to use a one-tailed or a two-tailed probability before looking at the data.
One-tailed test, as the name suggest is the statistical hypothesis test, in which the alternative hypothesis has a single end. On the other hand, two-tailed test implies the hypothesis test; wherein the alternative hypothesis has dual ends. In the one-tailed test, the alternative hypothesis is represented directionally.
HYPOTHESIS TESTING. A clinical trial begins with an assumption or belief, and then proceeds to either prove or disprove this assumption. In statistical terms, this belief or assumption is known as a hypothesis. Counterintuitively, what the researcher believes in (or is trying to prove) is called the "alternate" hypothesis, and the opposite ...
For a right-tailed test: For a left-tailed test: Test Statistic: Depending on the type of test and the distribution, the test statistic is computed (Z-score for normal distribution). Decision Rule: If the test statistic falls in the critical region, reject the null hypothesis in favor of the alternative hypothesis. Example: Effect of participants of students in coding competition on their fear ...
Two tailed test. The right tailed test and the left tailed test are examples of one-tailed tests. They are called "one tailed" tests because the rejection region (the area where you would reject the null hypothesis) is only in one tail. The two tailed test is called a two tailed test because the rejection region can be in either tail.
When delving into Business Intelligence (BI), hypothesis testing is a cornerstone of data analysis, providing insights and guiding decision-making. Choosing between a one-tailed and two-tailed ...
In a one-tailed test, the entire significance level is allocated to one tail of the distribution. For example, if you are using a significance level of 0.05 (5%), you would reject the null hypothesis if your data point falls in the 5% tail on either the right (for a right-tailed test) or the left (for a left-tailed test) end of the distribution.
The formula for the two-sample t-test can be written as: Where: X 1 and X 2 are the sample means of the two groups. s 1 and s 2 are the sample variances of the two groups. n 1 and n 2 are the sample sizes of the two groups. We then need to calculate the p-value using degrees of freedom equal to (n 1 +n 2 -1).
One-Tailed Test: A one-tailed test is a statistical test in which the critical area of a distribution is one-sided so that it is either greater than or less than a certain value, but not both. If ...
Hypothesis testing is the branch of statistical inference wherein you collect samples from a population to test an assumption about the population. In this case, the null hypothesis is that the true proportion of hands with North as declarer is 25%. The alternative hypothesis—based on our observation—is that North actually declares less ...