Test the claim about the population mean $\mu$ at the given level of significance. Assume the population is normally distributed. Find the $P$-value and determine whether you should reject or fail to reject the null hypothesis.
Claim: $\mu >52$, $\alpha =0.10$
Sample: $\bar{x}=53.1$, $s=4.7,$ $n=20$

• A. P-val = 0.154; Reject the null hypothesis.
• B. P-val = 0.154; Fail to reject the null hypothesis.
• C. P-val = 0.308; Reject the null hypothesis.
• D. P-val = 0.308; Fail to reject the null hypothesis.

Answer: (B)

Verified step by step guidance

1. Step 1: Identify the null hypothesis (H₀) and the alternative hypothesis (H₁). The claim is μ > 52, so the null hypothesis is H₀: μ ≤ 52, and the alternative hypothesis is H₁: μ > 52. This is a one-tailed test.
2. Step 2: Determine the test statistic formula. Since the population standard deviation is not provided and the sample size is small (n = 20), use the t-test formula: t = (x̄ - μ₀) / (s / √n), where x̄ is the sample mean, μ₀ is the hypothesized population mean, s is the sample standard deviation, and n is the sample size.
3. Step 3: Calculate the degrees of freedom (df) for the t-distribution. The degrees of freedom are given by df = n - 1. In this case, df = 20 - 1 = 19.
4. Step 4: Use the calculated t-value and the degrees of freedom to find the P-value. The P-value represents the probability of observing a test statistic as extreme as, or more extreme than, the calculated t-value under the null hypothesis. Since this is a one-tailed test, the P-value corresponds to the area to the right of the calculated t-value in the t-distribution.
5. Step 5: Compare the P-value to the significance level (α = 0.10). If the P-value is less than α, reject the null hypothesis. If the P-value is greater than or equal to α, fail to reject the null hypothesis. Based on the given P-value, determine the correct conclusion.