Problem
In Country A, the population mean height for 3-year-old boys is 39 inches. Suppose a random sample of 153 -year-old boys from Country B showed a sample mean of 38.4 inches with a standard deviation of 2 inches. The boys were independently sampled. Assume that heights are Normally distributed in the populati Complete parts a through c below. \[ t=-1.16 \] (Type an integer or decimal rounded to two decimal places as needed.) Find the $p$-value. \[ p=0.265 \] (Type an integer or decimal rounded to three decimal places as needed.) Reject or do not reject $\mathrm{H}_{0}$. Choose the correct answer below. A. Do not reject $\mathrm{H}_{0}$. The population mean is definitely 39 in. on the basis of these data at a significance level of 0.05 . B. Do not reject $\mathrm{H}_{0}$. There is no reason to believe that $39 \mathrm{in}$. is not the population mean at a significance level of 0.05 . C. Reject $\mathrm{H}_{0}$. There is reason to believe that 39 in. is not the population mean at a significance level of 0.05 . D. Reject $\mathrm{H}_{0}$. The population mean is definitely not 39 in. on the basis of these data at a significance level of 0.05 . b. Now suppose the sample consists of 30 boys instead of 15 and repeat the test. Find the test statistic. \[ t=\square \] (Type an integer or decimal rounded to two decimal places as needed.) Get more help Clear all
Solution
Step 1 ：The question is asking to find the test statistic for a sample of 30 boys. The test statistic in a t-test is calculated using the formula: \( t = \frac{\bar{x} - \mu}{s / \sqrt{n}} \) where: \(\bar{x}\) is the sample mean, \(\mu\) is the population mean, \(s\) is the standard deviation of the sample, and \(n\) is the sample size.
Step 2 ：In this case, we have: \(\bar{x} = 38.4\) inches, \(\mu = 39\) inches, \(s = 2\) inches, and \(n = 30\).
Step 3 ：We can substitute these values into the formula to find the test statistic.
Step 4 ：After executing this calculation, we get the value of the test statistic as \(t = -1.6431676725155024\).
Step 5 ：Final Answer: The test statistic is \(\boxed{-1.64}\).
