9. Hypothesis Testing for One Sample

Steps in Hypothesis Testing

9. Hypothesis Testing for One Sample

Steps in Hypothesis Testing

3:51 minutes

Problem 13.2.11

Textbook Question

In Exercises 9–12, use the sign test for the claim involving nominal data.

Overtime Rule in Football Before the overtime rule in the National Football League was changed in 2011, among 460 overtime games, 252 were won by the team that won the coin toss at the beginning of overtime. Using a 0.05 significance level, test the claim that the coin toss is fair in the sense that neither team has an advantage by winning it. Does the coin toss appear to be fair?

Verified step by step guidance

Identify the null and alternative hypotheses. The null hypothesis (H₀) is that the coin toss is fair, meaning the probability of winning after winning the coin toss is 0.5. The alternative hypothesis (H₁) is that the coin toss is not fair, meaning the probability of winning after winning the coin toss is not 0.5.

Determine the test statistic for the sign test. In this case, the number of successes (games won by the team that won the coin toss) is 252, and the total number of trials (overtime games) is 460. Under the null hypothesis, the expected number of successes is 460 × 0.5 = 230.

Calculate the test statistic. The test statistic for the sign test is based on the binomial distribution. Use the formula for the z-score: z = (x - np) / sqrt(np(1-p)), where x is the observed number of successes, n is the total number of trials, and p is the probability under the null hypothesis (0.5).

Determine the critical value or p-value. Since the significance level is 0.05, use a two-tailed test to find the critical z-values or calculate the p-value corresponding to the observed z-score. Compare the p-value to the significance level to decide whether to reject the null hypothesis.

Make a conclusion. If the p-value is less than 0.05 or the test statistic falls outside the critical region, reject the null hypothesis and conclude that the coin toss is not fair. Otherwise, fail to reject the null hypothesis and conclude that there is not enough evidence to suggest the coin toss is unfair.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above

Video duration:

Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Sign Test

The sign test is a non-parametric statistical method used to evaluate the median of a single sample or to compare two related samples. It is particularly useful for nominal data, where the outcomes can be categorized but not ranked. In this context, it helps determine if there is a significant difference in the outcomes of games won by the team that won the coin toss versus those that did not.

Null Hypothesis and Significance Level

In hypothesis testing, the null hypothesis represents a statement of no effect or no difference, which researchers aim to test against. The significance level, often denoted as alpha (α), is the threshold for determining whether to reject the null hypothesis. In this case, a significance level of 0.05 indicates that there is a 5% risk of concluding that a difference exists when there is none.

Nominal Data

Nominal data is a type of categorical data that represents distinct categories without any inherent order or ranking. Examples include gender, race, or, in this case, the outcome of a coin toss (win or lose). Understanding nominal data is crucial for applying the sign test, as it focuses on the frequency of occurrences within these categories rather than their numerical values.

Watch next

Master Step 1: Write Hypotheses with a bite sized video explanation from Patrick

Start learning

Related Videos

Related Practice

Textbook Question

Cola Weights Data Set 37 “Cola Weights and Volumes” in Appendix B lists the weights (lb) of the contents of cans of cola from four different samples: (1) regular Coke, (2) Diet Coke, (3) regular Pepsi, and (4) Diet Pepsi. The results from analysis of variance are shown in the Minitab display below. What is the null hypothesis for this analysis of variance test? Based on the displayed results, what should you conclude about H_knot. What do you conclude about equality of the mean weights from the four samples?

Textbook Question

Cola Weights For the four samples described in Exercise 1, the sample of regular Coke has a mean weight of 0.81682 lb, the sample of Diet Coke has a mean weight of 0.78479 lb, the sample of regular Pepsi has a mean weight of 0.82410 lb, and the sample of Diet Pepsi has a mean weight of 0.78386 lb. If we use analysis of variance and reach a conclusion to reject equality of the four sample means, can we then conclude that any of the specific samples have means that are significantly different from the others?

Textbook Question

Cola Weights The displayed results from Exercise 1 are from one-way analysis of variance. What is it about this test that characterizes it as one-way analysis of variance instead of two-way analysis of variance?

Textbook Question

In Exercises 9–12, use the sign test for the claim involving nominal data.

Medical Malpractice In a study of 1228 randomly selected medical malpractice lawsuits, it was found that 856 of them were dropped or dismissed (based on data from the Physicians Insurers Association of America). Use a 0.01 significance level to test the claim that there is a difference between the rate of medical malpractice lawsuits that go to trial and the rate of such lawsuits that are dropped or dismissed.

Textbook Question

In Exercises 13–16, refer to the indicated data set in Appendix B and use the sign test for the claim about the median of a population.

Cotinine in Smokers Data Set 15 “Passive and Active Smoke” includes cotinine measurements from 902 smokers. Cotinine is a biomarker of nicotine in the body. Use a 0.01 significance level to test the claim that smokers have cotinine levels with a median of 2.84 ng/mL, which is the median for nonsmokers not exposed to tobacco smoke.

Textbook Question

Hospital Admissions For the matched pairs listed in Exercise 1, identify the following components used in the Wilcoxon signed-ranks test:

a. Differences d

Textbook Question

Hospital Admissions For the matched pairs listed in Exercise 1, identify the following components used in the Wilcoxon signed-ranks test:

b. The ranks corresponding to the nonzero values of |d|

Textbook Question

Hospital Admissions For the matched pairs listed in Exercise 1, identify the following components used in the Wilcoxon signed-ranks test:

c. The signed ranks

Show more practices