In Exercises 29–32, test the claim about the population mean at the level of significance α. Assume the population is normally distributed.


Claim: ; μ ≤ 22,500; α = 0.01; α = 1200
Sample statistics: x_bar = 23,500, n = 45

Verified step by step guidance

1. Step 1: Identify the null hypothesis (H₀) and the alternative hypothesis (Hₐ). The claim is μ ≤ 22,500, so the null hypothesis is H₀: μ ≤ 22,500, and the alternative hypothesis is Hₐ: μ > 22,500. This is a right-tailed test.
2. Step 2: Calculate the test statistic using the formula for a one-sample z-test: z = (x̄ - μ₀) / (σ / √n), where x̄ is the sample mean, μ₀ is the population mean under the null hypothesis, σ is the population standard deviation, and n is the sample size.
3. Step 3: Substitute the given values into the formula. Here, x̄ = 23,500, μ₀ = 22,500, σ = 1200, and n = 45. Compute the standard error (SE) first: SE = σ / √n.
4. Step 4: Determine the critical value for a right-tailed test at α = 0.01. Use a z-table or statistical software to find the z-critical value corresponding to a significance level of 0.01.
5. Step 5: Compare the calculated z-test statistic to the critical value. If the test statistic is greater than the critical value, reject the null hypothesis H₀. Otherwise, fail to reject H₀. Interpret the result in the context of the claim.

Key Concepts

Hypothesis Testing

Hypothesis testing is a statistical method used to make decisions about a population based on sample data. It involves formulating a null hypothesis (H0) and an alternative hypothesis (H1), then using sample statistics to determine whether to reject H0 in favor of H1. In this case, the null hypothesis is that the population mean is less than or equal to 22,500.

Level of Significance (α)

The level of significance, denoted as α, is the probability of rejecting the null hypothesis when it is actually true, also known as a Type I error. In this scenario, α is set at 0.01, indicating a 1% risk of concluding that the population mean exceeds 22,500 when it does not. This threshold helps determine the critical value for making decisions based on the sample data.

Sample Mean and Standard Error

The sample mean (x̄) is the average value calculated from the sample data, which in this case is 23,500. The standard error (SE) measures the variability of the sample mean and is calculated as the population standard deviation divided by the square root of the sample size (n). Understanding these concepts is crucial for calculating test statistics and making inferences about the population mean.