Bundle: Statistics for the Behavioral Sciences, Loose-leaf Version, 10th + MindTap Psychology, 1 term (6 months) Printed Access Card
Bundle: Statistics for the Behavioral Sciences, Loose-leaf Version, 10th + MindTap Psychology, 1 term (6 months) Printed Access Card
10th Edition
ISBN: 9781337128995
Author: Frederick J Gravetter, Larry B. Wallnau
Publisher: Cengage Learning
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Chapter 8, Problem 13P

A random sample is selected from a normal population with a mean of μ = 100 and a standard deviation of σ = 20 . After a treatment is administered to the individuals in the sample, the sample mean is found to be M = 96 .

a. How large a sample is necessary for this sample mean to be statistically significant? Assume a two-tailed test with α = .05 .

b. If the sample mean were M = 98 , what sample size is needed to be significant for a two-tailed test with α = .05 ?

Expert Solution & Answer
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To determine
  1. How large a sample is necessary for the sample mean to be statistically significant if the sample mean =96?
  2. How large a sample is needed for the sample mean to be statistically significant if the sample mean =98 ?

Answer to Problem 13P

Solution:

  1. The sample size required for the sample mean =96 to be statistically significant is 97.
  2. The sample size required for the sample mean =98 to be statistically significant is 385.

Explanation:

  1. To determine a sample size that is necessary for the sample mean =96 to be statistically significant, we will make use of z statistic.

    We are given:

    μ=100, σ=20, M=96 and n=?

    We know that the two-tailed critical value at 5% significance level =±1.96. Now in order to determine the sample size for the sample mean to be statistically significant we will have

    z - score =±1.96. Therefore, we have:

    z=Mμσn

    Let's take z=1.96

    1.96=9610020n

    1.96=4×n20

    1.96×20=4×n

    39.2=4×n

    n=39.24

    n=9.8

    n=9.82=96.0497

    Therefore, the sample size required for the sample mean =96 to be statistically significant is 97.

  2. To determine a sample size that is necessary for the sample mean =98 to be statistically significant, we will make use of z statistic.

    We are given:

    μ=100, σ=20, M=98 and n=?

    We know that the two-tailed critical value at 5% significance level =±1.96. Now in order to determine the sample size for the sample mean to be statistically significant we will have

    z - score =±1.96. Therefore, we have:

    z=Mμσn

    Let's take z=1.96

    1.96=9810020n

    1.96=2×n20

    1.96×20=2×n

    39.2=2×n

    n=39.22

    n=19.6

    n=19.62=384.16385

    Therefore, the sample size required for the sample mean =98 to be statistically significant is 385.

Conclusion:
  1. The sample size required for the sample mean =96to be statistically significant at α=0.05 is 97.
  2. The sample size required for the sample mean =98to be statistically significant at α=0.05 is 385.

Explanation of Solution

  1. To determine a sample size that is necessary for the sample mean =96 to be statistically significant, we will make use of z statistic.

    We are given:

    μ=100, σ=20, M=96 and n=?

    We know that the two-tailed critical value at 5% significance level =±1.96. Now in order to determine the sample size for the sample mean to be statistically significant we will have

    z - score =±1.96. Therefore, we have:

    z=Mμσn

    Let's take z=1.96

    1.96=9610020n

    1.96=4×n20

    1.96×20=4×n

    39.2=4×n

    n=39.24

    n=9.8

    n=9.82=96.0497

    Therefore, the sample size required for the sample mean =96 to be statistically significant is 97.

  2. To determine a sample size that is necessary for the sample mean =98 to be statistically significant, we will make use of z statistic.

    We are given:

    μ=100, σ=20, M=98 and n=?

    We know that the two-tailed critical value at 5% significance level =±1.96. Now in order to determine the sample size for the sample mean to be statistically significant we will have

    z - score =±1.96. Therefore, we have:

    z=Mμσn

    Let's take z=1.96

    1.96=9810020n

    1.96=2×n20

    1.96×20=2×n

    39.2=2×n

    n=39.22

    n=19.6

    n=19.62=384.16385

    Therefore, the sample size required for the sample mean =98 to be statistically significant is 385.

Conclusion:
  1. The sample size required for the sample mean =96to be statistically significant at α=0.05 is 97.
  2. The sample size required for the sample mean =98to be statistically significant at α=0.05 is 385.

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