Questions: To test H0: μ=100 versus H1: μ ≠ 100, a simple random sample of size n=24 is obtained from a population that is known to be normally distributed. Answer parts (a)-(e). (c) Draw a t-distribution that depicts the critical region(s). Which of the following graphs shows the critical region(s) in the t-distribution? A. B. C. (d) Will the researcher reject the null hypothesis? A. The researcher will reject the null hypothesis since the test statistic is not in the rejection region. B. The researcher will reject the null hypothesis since the test statistic is in the rejection region. C. There is not sufficient evidence for the researcher to reject the null hypothesis since the test statistic is not in the rejection region. D. There is not sufficient evidence for the researcher to reject the null hypothesis since the test statistic is in the rejection region. (e) Construct a 99% confidence interval to test the hypothesis. The confidence interval is ( ). (Round to two decimal places as needed.)

To test H0: μ=100 versus H1: μ ≠ 100, a simple random sample of size n=24 is obtained from a population that is known to be normally distributed. Answer parts (a)-(e). (c) Draw a t-distribution that depicts the critical region(s). Which of the following graphs shows the critical region(s) in the t-distribution? A. B. C. (d) Will the researcher reject the null hypothesis? A. The researcher will reject the null hypothesis since the test statistic is not in the rejection region. B. The researcher will reject the null hypothesis since the test statistic is in the rejection region. C. There is not sufficient evidence for the researcher to reject the null hypothesis since the test statistic is not in the rejection region. D. There is not sufficient evidence for the researcher to reject the null hypothesis since the test statistic is in the rejection region. (e) Construct a 99% confidence interval to test the hypothesis. The confidence interval is ( ). (Round to two decimal places as needed.)
Transcript text: To test $H_{0}: \mu=100$ versus $H_{1}: \mu \neq 100$, a simple random sample of size $n=24$ is obtained from a population that is known to be normally distributed. Answer parts (a)-(e). (c) Draw a t-distribution that depicts the critical region(s). Which of the following graphs shows the critical region(s) in the t-distribution? A. B. C. (d) Will the researcher reject the null hypothesis? A. The researcher will reject the null hypothesis since the test statistic is not in the rejection region. B. The researcher will reject the null hypothesis since the test statistic is in the rejection region. C. There is not sufficient evidence for the researcher to reject the null hypothesis since the test statistic is not in the rejection region. D. There is not sufficient evidence for the researcher to reject the null hypothesis since the test statistic is in the rejection region. (e) Construct a $99 \%$ confidence interval to test the hypothesis. The confidence interval is $\square$ $\square$ ). (Round to two decimal places as needed.)
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Solution

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Solution Steps

Step 1: Identify the Hypotheses
  • Null Hypothesis (H0): μ = 100
  • Alternative Hypothesis (H1): μ ≠ 100
Step 2: Determine the Distribution and Critical Regions
  • The sample size (n) is 24.
  • The population is normally distributed.
  • Use the t-distribution to determine the critical regions for a two-tailed test.
Step 3: Draw the t-Distribution and Identify Critical Regions
  • The critical regions are the areas in the tails of the t-distribution where the null hypothesis would be rejected.
  • For a two-tailed test with a 5% significance level, each tail will have 2.5% of the distribution.
  • The correct graph is the one that shows the critical regions in both tails of the distribution.

Final Answer

  • The correct graph is the one that shows the critical regions in both tails of the t-distribution.
  • The researcher will reject the null hypothesis if the test statistic falls in either of the critical regions.
  • Construct a 99% confidence interval using the t-distribution with n-1 degrees of freedom.
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