Question

Find the standardized test statistic for testing a claim H0: μ ≤ 26.9 at significance α = 0.05 given a sample with n = 36 and begin mathsize 14px style x with bar on top end style = 27.9, if the population is normally distributed and σ = 10.56. Round your answer to three decimal places.

Answer 1

To calculate the standardized test statistic for the hypothesis test H0: μ ≤ 26.9 with a sample mean of 27.9, sample size of 36, and population standard deviation of 10.56, the z-score is determined using the formula and given values, resulting in a z-score of 0.568.

Step-by-step explanation:

To find the standardized test statistic (often referred to as the z-score) for testing the hypothesis H0: μ ≤ 26.9 against the alternative hypothesis Ha: μ > 26.9 using a significance level α = 0.05, we will use the given sample statistics (sample mean ¯x = 27.9, sample size n = 36, and population standard deviation σ = 10.56). Since the population standard deviation is known and the sample size is greater than 30, we can use the z-distribution to perform the hypothesis test.

The formula to calculate the z-score is:

z = (¯x - μ0) / (σ / √n)

Substituting in the given values:

z = (27.9 - 26.9) / (10.56 / √36)

z = 1.0 / (10.56 / 6)

z = 1.0 / (1.76)

z = 0.568

Round this value to three decimal places, the standardized test statistic is 0.568.