Question

4-2.5 In a class of 50 students, the result of a particular examination is a true mean of 70 and a true variance of 12. It is desired to estimate the mean by sampling, without replacement, a subset of the scores.

a) Find the standard deviation of the sample mean if only 10 scores are used.

b) How large should the sample size be for the standard deviation of the sample mean to be one percentage point (out of 100)?

c) How large should the sample size be for the standard deviation of the sample mean to be 1 % of the true mean?

I don't think need more informations in case this q from (Cooper - Probabilistic Methods of Signal and System Analysis, 3rd Ed) page 185

Answer 1

We are given with:

Sample size, n = 50 students

True mean, µ = 70

True variance, σ² = 12

a) Find the standard deviation of the sample mean if only 10 scores are used.

If only 10 scores are used, then sample size, n = 10 students

We know that:

Standard deviation , σ = √Variance = √σ²

σ = √12

Thus, Standard deviation of the sample mean, σ = σ/ √n

σ = √12/ √10

σ = 1.095445

Thus, Standard deviation of the sample mean is 1.095445.

b) How large should the sample size be for the standard deviation of the sample mean to be one percentage point (out of 100)?

We are given that sample standard deviation, s = 1

So, Sample size, n = (σ/s)² = (√12/1)² = 12

Therefore, the sample size for the standard deviation of the sample mean to be one percentage point is 12.

c) How large should the sample size be for the standard deviation of the sample mean to be 1 % of the true mean?

We are given that sample standard deviation, s = 1% of the true mean = 0.01*70 = 0.7

So, Sample size, n = (σ/s)² = (√12/0.7)² = 24.4898 = 25

Therefore, the sample size for the standard deviation of the sample mean to be 1 % of the true mean is 25.