Table of contents

1. Intro to Stats and Collecting Data55m
2. Describing Data with Tables and Graphs1h 55m
3. Describing Data Numerically1h 45m
4. Probability2h 16m
5. Binomial Distribution & Discrete Random Variables2h 33m
6. Normal Distribution and Continuous Random Variables1h 38m
7. Sampling Distributions & Confidence Intervals: Mean1h 53m
8. Sampling Distributions & Confidence Intervals: Proportion1h 12m
- Sampling Distribution of Sample Proportion
  29m
- Confidence Intervals for Population Proportion
  42m
9. Hypothesis Testing for One Sample2h 19m
10. Hypothesis Testing for Two Samples3h 22m
11. Correlation1h 6m
12. Regression1h 4m
13. Chi-Square Tests & Goodness of Fit1h 20m
14. ANOVA1h 0m

9. Hypothesis Testing for One Sample

Steps in Hypothesis Testing

Statistics

9. Hypothesis Testing for One Sample

Steps in Hypothesis Testing

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2:09 minutes

Problem 7.RE.29

Textbook Question

In Exercises 29–34, find the critical value(s) and rejection region(s) for the type of t-test with level of significance α and sample size n.

Two-tailed test, α=0.05, n=20

Verified step by step guidance

Determine the degrees of freedom (df) for the t-test. The formula for degrees of freedom is df = n - 1, where n is the sample size. In this case, n = 20, so df = 20 - 1.

Identify the level of significance (α) for the test. For a two-tailed test, the significance level is split equally between the two tails of the t-distribution. Here, α = 0.05, so each tail will have α/2 = 0.025.

Use a t-distribution table or statistical software to find the critical t-value(s) corresponding to df = 19 and a cumulative probability of 1 - α/2 = 0.975 for the upper tail and α/2 = 0.025 for the lower tail.

The rejection region(s) for the two-tailed test are defined as the areas in the tails of the t-distribution where the test statistic falls beyond the critical t-values. Specifically, the rejection regions are t < -t_critical and t > t_critical.

Summarize the critical values and rejection regions. For example, if the critical t-values are ±t_critical, the rejection regions are t < -t_critical and t > t_critical. These regions indicate where the null hypothesis would be rejected.

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Key Concepts

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

Critical Value

A critical value is a point on the scale of the test statistic beyond which we reject the null hypothesis. In a two-tailed test, there are two critical values that correspond to the significance level (α). For example, with α = 0.05, the critical values define the boundaries of the rejection regions in both tails of the distribution.

Rejection Region

The rejection region is the area in the tails of the probability distribution where we reject the null hypothesis. For a two-tailed test with α = 0.05, the rejection regions are located in both tails, each containing 2.5% of the total probability. If the test statistic falls within these regions, we conclude that the observed data is statistically significant.

t-Test

A t-test is a statistical test used to determine if there is a significant difference between the means of two groups, especially when the sample size is small (n < 30) and the population standard deviation is unknown. The t-test uses the t-distribution, which accounts for the additional uncertainty introduced by estimating the population standard deviation from the sample.

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In Exercises 29–34, find the critical value(s) and rejection region(s) for the type of t-test with level of significance α and sample size n.Two-tailed test, α=0.05, n=20

Key Concepts

Critical Value

Rejection Region

t-Test

Watch next

In Exercises 29–34, find the critical value(s) and rejection region(s) for the type of t-test with level of significance α and sample size n.

Two-tailed test, α=0.05, n=20