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: $\mu \ne 1020$, $\alpha =0.01$, $\sigma =85$
Sample: $\bar{x}=990$, $n=40$

• A. P-val: 0.026; Reject the null hypothesis
• B. P-val: 0.026; Fail to reject the null hypothesis
• C. P-val: 0.013; Reject the null hypothesis
• D. P-val: 0.013; 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 null hypothesis is H₀: μ = 1020, and the alternative hypothesis is H₁: μ ≠ 1020. This is a two-tailed test because the claim is that the population mean is not equal to 1020.
2. Step 2: Calculate the test statistic using the formula for the z-test: z = (x̄ - μ) / (σ / √n), where x̄ is the sample mean, μ is the population mean under the null hypothesis, σ is the population standard deviation, and n is the sample size. Substitute the given values: x̄ = 990, μ = 1020, σ = 85, and n = 40.
3. Step 3: Determine the P-value for the calculated z-test statistic. Since this is a two-tailed test, the P-value is found by doubling the area in the tails of the standard normal distribution corresponding to the absolute value of the z-test statistic.
4. Step 4: Compare the P-value to the significance level α = 0.01. If the P-value is less than α, reject the null hypothesis. Otherwise, fail to reject the null hypothesis.
5. Step 5: Based on the comparison, make a decision about the null hypothesis and interpret the result in the context of the problem. If the null hypothesis is rejected, it means there is sufficient evidence to support the claim that the population mean is not equal to 1020. If the null hypothesis is not rejected, it means there is insufficient evidence to support the claim.