9. Hypothesis Testing for One Sample

Steps in Hypothesis Testing

9. Hypothesis Testing for One Sample

Steps in Hypothesis Testing

3:06 minutes

Problem 12.CQ.1

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?

Verified step by step guidance

Step 1: Understand the null hypothesis (H₀) for an analysis of variance (ANOVA) test. The null hypothesis states that the mean weights of the contents of the cans from the four samples (regular Coke, Diet Coke, regular Pepsi, Diet Pepsi) are equal. Mathematically, H₀: μ₁ = μ₂ = μ₃ = μ₄, where μ represents the mean weight for each sample.

Step 2: Examine the Minitab output provided. Focus on the P-Value column, which is used to test the null hypothesis. The P-Value for the 'Factor' source is 0.000, which is significantly less than the common significance level of 0.05.

Step 3: Interpret the P-Value. Since the P-Value is less than 0.05, we reject the null hypothesis (H₀). This indicates that there is sufficient evidence to conclude that at least one of the mean weights is different from the others.

Step 4: Analyze the F-Value. The F-Value is 503.06, which is very large. A high F-Value further supports the conclusion that the variation between the group means is significant compared to the variation within the groups.

Step 5: Conclude about the equality of mean weights. Based on the results, we conclude that the mean weights of the contents of the cans from the four samples are not equal. There is a statistically significant difference in the mean weights among the four groups.

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.

Null Hypothesis (H0)

The null hypothesis (H0) is a statement that there is no effect or no difference, and it serves as a starting point for statistical testing. In the context of the ANOVA test for the cola weights, H0 posits that the mean weights of the four cola samples (regular Coke, Diet Coke, regular Pepsi, and Diet Pepsi) are equal. Rejecting H0 suggests that at least one sample mean is different from the others.

Analysis of Variance (ANOVA)

ANOVA is a statistical method used to compare the means of three or more groups to determine if at least one group mean is significantly different from the others. It assesses the variance within each group and between groups, providing an F-value that indicates the ratio of these variances. A significant F-value, as shown in the Minitab output, suggests that the group means are not all equal.

P-Value

The P-value is a statistical measure that helps determine the significance of the results obtained from a hypothesis test. It represents the probability of observing the data, or something more extreme, assuming the null hypothesis is true. In the provided Minitab output, a P-value of 0.000 indicates strong evidence against the null hypothesis, leading to the conclusion that there are significant differences in the mean weights of the cola samples.

Watch next

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

Start learning

Related Videos

Related Practice

Textbook Question

Using the Kruskal-Wallis Test

In Exercises 5–8, use the Kruskal-Wallis test.

HIC Measurements Use the sample data from Exercise 1 with a 0.05 significance level to test the claim that small, midsize, large, and SUV vehicles have the same median HIC measurement in car crash tests.

Textbook Question

Arsenic in Rice Listed below are amounts of arsenic in samples of brown rice from three different states. The amounts are in micrograms of arsenic and all samples have the same serving size. The data are from the Food and Drug Administration. Use a 0.01 significance level to test the claim that the three samples are from populations with the same median.

Textbook Question

Clancy, Rowling, and Tolstoy Ease of Reading Pages were randomly selected from three books: The Bear and the Dragon by Tom Clancy, Harry Potter and the Sorcerer’s Stone by J. K. Rowling, and War and Peace by Leo Tolstoy. Listed below are Flesch Reading Ease Scores for those pages. Higher scores correspond to pages that are easier to read. Use a 0.01 significance level to test the claim that pages from books by those three authors have the same median Flesch Reading Ease score.

Textbook Question

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

e. The value of the test statistic T

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 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?

Show more practices