Question

Suppose that you are testing the hypotheses H0​: u = 84 vs. HA​: u ≠ 84. A sample of size 61 results in a sample mean of 89 and a sample standard deviation of 1.4

A) What is the standard error of the​ mean?
B) What is the critical value of​ t* for a 95% confidence ​interval? ​
C) Construct a 95% confidence interval for μ. ​
D) Based on the confidence​ interval, at α=0.050 can you reject H0​? Explain.

Answer 1

The standard error of the mean is approximately 0.179, the critical value of t* for a 95% confidence interval is approximately 2.00, and the 95% confidence interval for μ is (88.642, 89.358). The null hypothesis H0: μ = 84 can be rejected at α=0.050 since 84 is not within the CI.

Step-by-step explanation:

Testing Hypotheses and Confidence Intervals

A) Standard Error of the Mean: The standard error (SE) of the mean is calculated as the sample standard deviation divided by the square root of the sample size (n). Using the given data (sample standard deviation of 1.4 and n=61), the SE is 1.4 / √61 ≈ 0.179.

B) Critical Value of t*: For a 95% confidence interval with df=n-1 (degrees of freedom), where n=61, we refer to the t-distribution table or use statistical software to determine the critical value. For df=60, the t* value at the 95% confidence level is approximately 2.00.

C) 95% Confidence Interval for μ: The confidence interval (CI) is calculated using the formula: sample mean ± (t* × SE). So, the CI for μ is 89 ± (2.00 × 0.179), which gives us the interval (88.642, 89.358).

D) Hypothesis Test Conclusion: The null hypothesis H0 ​: μ = 84 is tested against the alternative hypothesis HA ​: μ ≠ 84. Since the null hypothesis value of 84 is not contained within the 95% CI of (88.642, 89.358), we reject H0 at α=0.050.