Question

A 95onfidence interval for a population mean was reported to be 152 to 160. If the population standard deviation is equal to 15, what sample size was used in this study?

Answer 1

The sample size used in the study is 54 approximately.

In the question, we are given that a 95% confidence interval for a population mean was reported to be 152 to 160.

We are asked to find the sample size that would have been studied if the population standard deviation is equal to 15.

The confidence level is given = 95%.

The Z-Score that corresponds to the given confidence level = 1.96.

The given standard deviation, σ = 15.

The confidence interval is given to be 152 to 160, that is, the Upper confidence limit, UCL = 160, and the lower confidence limit, LCL = 152.

We assume the sample size to be n.

The Margin of error, MOE, can be given by two formulas:

1. MOE = (UCL - LCL)/2,
2. MOE = Zσ/√n.

Equating the two formulas, we get:

(UCL - LCL)/2 = Zσ/√n,

or, √n = (2Zσ)/(UCL - LCL),

or, n = {(2Zσ)/(UCL - LCL)}²,

or, n = {(2*1.96*15)/(160 - 152)}²,

or, n = (58.8/8)²,

or, n = 7.35²,

or, n = 54.0225,

or, n ≈ 54.

Thus, the sample size used in the study is 54 approximately.

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