Question

The physical fitness score for a population of police officers at a local police station is 72 , with a standard deviation of 7 on a 100 -point physical endurance scale. Suppose the police chief selects a sample of 49 local police officers from this population and records a mean physical fitness rating on this scale equal to 74. He conducts a one-sample z test to determine whether physical endurance increased at a .05 level of significance. a. State the value of the test statistic and whether to retain or reject the null hypothesis.

Answer 1

The test statistic is 2, calculated using the formula Z = (X - μ) / (σ / √n). Since this value exceeds the critical z-score of 1.96 for a 0.05 level of significance in a two-tailed test, we reject the null hypothesis, suggesting an increase in mean physical endurance.

Step-by-step explanation:

The student is asking how to calculate the test statistic for a one-sample z test and to determine whether to retain or reject the null hypothesis at a 0.05 level of significance.

To calculate the test statistic, we use the formula:

Z = (X - μ) / (σ / √n)

Where X is the sample mean, μ is the population mean, σ is the population standard deviation, and n is the sample size.

Given: X = 74, μ = 72, σ = 7, and n = 49.

Plugging in these values:

Z = (74 - 72) / (7 / √49) = 2 / (7 / 7) = 2

The z-score for a 0.05 level of significance for a two-tailed test is approximately ±1.96. Since 2 > 1.96, we reject the null hypothesis. This suggests that there is significant evidence at the 5% level of significance that the mean physical endurance has increased.