7. A normal population has a mean of = 100. A sample of N = 36 is selected from

Question

7. A normal population has a mean of = 100. A sample of N = 36 is selected from the population, and a treatment is administered to the sample. After treatment, the sample mean is computed to be sample mean = 106.

a. Assuming that the population standard deviation is = 12, use the data to test whether or not the treatment has a significant effect. Do a z test with = .05

b. Repeat the z-Test, but this time assume that the population standard deviation is = 30.

c. Compare the results from part a and part b. How does the population standard deviation influence the outcome of a z-test?

Explanation / Answer

7. A normal population has a mean of = 100. A sample of N = 36 is selected from the population, and a treatment is administered to the sample. After treatment, the sample mean is computed to be sample mean = 106.

a. Assuming that the population standard deviation is = 12, use the data to test whether or not the treatment has a significant effect. Do a z test with = .05

Solution:

The null and alternative hypothesis for this test is given as below:

H0: µ = 100 versus Ha: µ 100

Level of significance = alpha = 0.05

Test statistic formula is given as below:

Z = (Xbar - µ) / (/sqrt(n)]

Z = (106 – 100) / [12/sqrt(36)]

Z = 6/[12/6] = 6/2 = 3

P-value = 0.0027

Alpha value = 0.05

P-value < Alpha value

So, we reject the null hypothesis

We conclude that there is sufficient evidence that the treatment has a significant effect.

b. Repeat the z-Test, but this time assume that the population standard deviation is = 30.

Solution:

The null and alternative hypothesis for this test is given as below:

H0: µ = 100 versus Ha: µ 100

Level of significance = alpha = 0.05

Test statistic formula is given as below:

Z = (Xbar - µ) / (/sqrt(n)]

Z = (106 – 100) / [30/sqrt(36)]

Z = 6/[30/6] = 6/5 = 1.20

P-value = 0.2301

Alpha value = 0.05

P-value > Alpha value

So, we do not reject the null hypothesis

We conclude that there is no sufficient evidence that the treatment has a significant effect.

c. Compare the results from part a and part b. How does the population standard deviation influence the outcome of a z-test?

Solution:

If the value of the population standard deviation increases, the result of the z test is changes from the rejection of the null hypothesis to failing of rejection of null hypothesis. So, the value of the population standard deviation is very important for deciding whether or not rejects the null hypothesis.

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