Final answer:
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.