A simple random sample of 100 postal employees is used to test if the average time postal employees have worked for the postal service has changed from the value of 7.5 years recorded 20 years ago.The sample mean was 0 :  = 7.5 versus H a :   7.5.A one-sample t test will be used.What is the P-value for the one-sample t test? A) P-value

Question

A simple random sample of 100 postal employees is used to test if the average time postal employees have worked for the postal service has changed from the value of 7.5 years recorded 20 years ago.The sample mean was

= 7 years with a standard deviation of s = 2 years.Assume the distribution of the time the employees have worked for the postal service is approximately Normal.The hypotheses being tested are H₀:  = 7.5 versus H_a:   7.5.A one-sample t test will be used.What is the P-value for the one-sample t test?

A) P-value < 0.01
B) 0.01 < P-value < 0.05
C) 0.05 < P-value < 0.10
D) P-value > 0.10

Candice Rimas · Accepted Answer

Final Answer : B, Explanation :  The test statistic for the one-sample t test is:t = (x̄ - μ) / (s / √n)where x̄ is the sample mean, μ is the population mean (the value being tested against), s is the sample standard deviation, and n is the sample size.In this case, the sample mean is x̄ = 7, the population mean is μ = 7.5 (the value being tested against), the sample standard deviation is s = 2, and the sample size is n = 100.Plugging in the numbers, we get:t = (7 - 7.5) / (2 / √100) = -2.5The degrees of freedom for the one-sample t test is n-1 = 99.Looking up the t-value in the t-table with 99 degrees of freedom, we get:tα/2,99 = 1.984 (for a two-tailed test with α = 0.05)Since the calculated t-value (-2.5) is less than the critical t-value (-1.984), we reject the null hypothesis and conclude that the average time postal employees have worked for the postal service is different from 7.5 years.The P-value for the test is the area in the t-distribution to the left of -2.5 (since it is a two-tailed test). Using a t-table or a t-distribution calculator, we find that the P-value is approximately 0.014.Therefore, the answer is B (0.01 < P-value < 0.05), meaning that the observed sample mean (7 years) is significantly different from the assumed population mean (7.5 years) at a 5% significance level.

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