Table of contents

1. Intro to Stats and Collecting Data1h 14m
2. Describing Data with Tables and Graphs1h 55m
3. Describing Data Numerically2h 5m
4. Probability2h 16m
5. Binomial Distribution & Discrete Random Variables3h 6m
6. Normal Distribution and Continuous Random Variables2h 11m
7. Sampling Distributions & Confidence Intervals: Mean3h 23m
8. Sampling Distributions & Confidence Intervals: Proportion1h 12m
- Sampling Distribution of Sample Proportion
  29m
- Confidence Intervals for Population Proportion
  42m
9. Hypothesis Testing for One Sample3h 29m
10. Hypothesis Testing for Two Samples4h 50m
11. Correlation1h 6m
12. Regression1h 50m
13. Chi-Square Tests & Goodness of Fit1h 57m
14. ANOVA1h 57m

9. Hypothesis Testing for One Sample

Steps in Hypothesis Testing

Statistics

9. Hypothesis Testing for One Sample

Steps in Hypothesis Testing

Previous problemNext problem

3:15 minutes

Problem 8.3.4

Textbook Question

Test Statistic and Critical Value The statistics for the sample data in Exercise 1 are n = 15, x_bar = 6.133333, and s = 8.862978, where the units are millions of dollars. Find the test statistic and critical value(s) for a test of the claim that the salaries are from a population with a mean greater than 5 million dollars. Assume that a 0.05 significance level is used.

Verified step by step guidance

Step 1: Identify the hypothesis. The null hypothesis (H₀) is that the population mean μ is equal to 5 million dollars (H₀: μ = 5). The alternative hypothesis (H₁) is that the population mean μ is greater than 5 million dollars (H₁: μ > 5). This is a one-tailed test.

Step 2: Calculate the test statistic using the formula for the t-test: t = (x̄ - μ) / (s / √n). Here, x̄ is the sample mean (6.133333), μ is the hypothesized population mean (5), s is the sample standard deviation (8.862978), and n is the sample size (15). Substitute the given values into the formula.

Step 3: Determine the degrees of freedom (df) for the t-test. The degrees of freedom for a single-sample t-test is calculated as df = n - 1. For this problem, df = 15 - 1 = 14.

Step 4: Find the critical value for the t-test at a significance level of 0.05 for a one-tailed test. Use a t-distribution table or statistical software to find the critical t-value corresponding to df = 14 and α = 0.05.

Step 5: Compare the calculated test statistic to the critical value. If the test statistic is greater than the critical value, reject the null hypothesis (H₀). Otherwise, fail to reject the null hypothesis.

Verified video answer for a similar problem:

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

Video duration:

Play a video:

Was this helpful?

Key Concepts

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

Test Statistic

A test statistic is a standardized value that is calculated from sample data during a hypothesis test. It measures how far the sample mean is from the null hypothesis mean, expressed in terms of standard errors. In this context, the test statistic will help determine whether the sample mean salary significantly exceeds the hypothesized population mean of 5 million dollars.

Critical Value

The critical value is a threshold that determines the boundary for rejecting the null hypothesis in a statistical test. It is derived from the significance level (alpha), which in this case is 0.05. For a one-tailed test, the critical value indicates the point beyond which the test statistic would lead to rejecting the null hypothesis, thus supporting the claim that the mean salary is greater than 5 million dollars.

Significance Level

The significance level, denoted as alpha, is the probability of rejecting the null hypothesis when it is actually true, also known as a Type I error. In this scenario, a significance level of 0.05 implies that there is a 5% risk of concluding that the mean salary is greater than 5 million dollars when it is not. This level helps to control the likelihood of making erroneous conclusions based on the sample data.

Watch next

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

Start learning

Related Videos

Related Practice

Textbook Question

RESAMPLING

a. In general, what does it mean to “resample” the following data set consisting of wait times (minutes) of customers waiting in line for the Space Mountain ride at Walt Disney World: 50, 25, 75, 35, 50?

Textbook Question

Using Confidence Intervals to Test Hypotheses When analyzing the last digits of telephone numbers in Port Jefferson, it is found that among 1000 randomly selected digits, 119 are zeros. If the digits are randomly selected, the proportion of zeros should be 0.1.

c. Use the sample data to construct a 95% confidence interval estimate of the proportion of zeros. What does the confidence interval suggest about the claim that the proportion of zeros equals 0.1?

Textbook Question

d. Compare the results from the critical value method, the P-value method, and the confidence interval method. Do they all lead to the same conclusion?

Textbook Question

Claim of “At Least” or “At Most”

How do the following results change?

a. Chapter Problem claim is changed to this: “At least 50% of Internet users utilize two-factor authentication to protect their online data.”

Textbook Question

Finding P-values

In Exercises 5–8, either use technology to find the P-value or use Table A-3 to find a range of values for the P-value. Based on the result, what is the final conclusion?

Weights of Quarters The claim is that weights (grams) of quarters made after 1964 have a mean equal to 5.670 g as required by mint specifications. The sample size is and the test statistic is t = -3.135.

Textbook Question

Finding P-values

In Exercises 5–8, either use technology to find the P-value or use Table A-3 to find a range of values for the P-value. Based on the result, what is the final conclusion?

Cotinine in Smokers The claim is that smokers have a mean cotinine level greater than the level of 2.84 ng/mL found for nonsmokers. (Cotinine is used as a biomarker for exposure to nicotine.) The sample size is n = 902 and the test statistic is t = 56.319.

Textbook Question

Hypothesis Test for Lightning Deaths Refer to the sample data given in Cumulative Review Exercise 1 and consider those data to be a random sample of annual lightning deaths from recent years. Use those data with a 0.01 significance level to test the claim that the mean number of annual lightning deaths is less than the mean of 72.6 deaths from the 1980s. If the mean is now lower than in the past, identify one of the several factors that could explain the decline.

Textbook Question

Testing Claims About Variation

In Exercises 5–16, test the given claim. Identify the null hypothesis, alternative hypothesis, test statistic, P-value, or critical value(s), then state the conclusion about the null hypothesis, as well as the final conclusion that addresses the original claim. Assume that a simple random sample is selected from a normally distributed population.

Birth Weights A simple random sample of birth weights of 30 girls has a standard deviation of 829.5 g. Use a 0.01 significance level to test the claim that birth weights of girls have the same standard deviation as birth weights of boys, which is 660.2 g (based on Data Set 6 “Births” in Appendix B).

Show more practices

Download the Mobile app

Do not sell my personal information

Accessibility

Patent notice

Sitemap