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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2:07 minutes

Problem 7.RE.34

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=0.02, n=12

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 = 12, so df = 12 - 1 = 11.

Identify the level of significance (α) for the two-tailed test. Since α = 0.02, divide it equally between the two tails. This means each tail will have an area of α/2 = 0.02/2 = 0.01.

Use a t-distribution table or statistical software to find the critical t-value(s) corresponding to df = 11 and a cumulative probability of 1 - α/2 = 1 - 0.01 = 0.99 for the upper tail. For the lower tail, the critical t-value will be the negative of the upper tail value.

Define the rejection region(s) based on the critical t-values. For a two-tailed test, the rejection regions are t < -t_critical and t > t_critical, where t_critical is the positive critical value obtained in the previous step.

Summarize the results: The critical t-values and rejection regions are determined, and these will guide the decision to reject or fail to reject the null hypothesis based on the test statistic.

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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 that separates the region where the null hypothesis is rejected from the region where it is not rejected. In hypothesis testing, critical values are determined based on the significance level (α) and the type of test being conducted, such as a t-test. For a two-tailed test, critical values are found at both tails of the distribution.

Rejection Region

The rejection region is the set of all values of the test statistic that would lead to the rejection of the null hypothesis. In a two-tailed test, this region is divided into two parts, corresponding to the critical values on either side of the distribution. The size of the rejection region is determined by the significance level (α), which indicates the probability of making a Type I error.

Two-Tailed Test

A two-tailed test is a statistical test that evaluates whether a sample mean is significantly different from a population mean in either direction (higher or lower). This type of test is used when the alternative hypothesis does not specify a direction of the effect. In the context of the given question, a two-tailed test with α=0.02 means that the rejection regions will be located in both tails of the t-distribution, each containing 1% of the total area.

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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=0.02, n=12

Key Concepts

Critical Value

Rejection Region

Two-Tailed 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=0.02, n=12