Question

The mean value of a random sample of 60 items was found to be 145, with a standard deviation of 40. Find the 95% confidence limits for the population mean. What size of the sample is required to estimate the population mean within 5 of its actual value with 95% or more confidence, using the sample mean? a) 50 b) 75 c) 100 d) 125

Answer 1

Option (a) is correct because a sample size of 50 is sufficient to estimate the population mean within 5 of its actual value with 95% confidence, based on the given conditions and calculations.

Step-by-step explanation:

In order to calculate the 95% confidence limits for the population mean, we use the formula for the confidence interval:

`\[ \bar{x} \pm Z \left((s)/(√(n))\right) \]`

Where:

• is the sample mean,

• s is the standard deviation of the sample,

• n is the sample size, and

• Z is the Z-score associated with the desired confidence level.

Given in the question,
`\(\bar{x} = 145\), \(s = 40\)`, and
`\(n = 60\)`. The Z-score for a 95% confidence interval is approximately 1.96.

`\[ 145 \pm 1.96 \left((40)/(√(60))\right) \]`

Calculating this, we get the confidence interval for the population mean.

Now, to find the required sample size to estimate the population mean within 5 of its actual value with 95% confidence, we use the formula:

`\[ n = \left((Z * s)/(E)\right)^2 \]`

Where:

• E is the margin of error (5 in this case).

Substituting the given values, we get:

`\[ n = \left((1.96 * 40)/(5)\right)^2 \]`

Calculating this, we find that the required sample size is approximately 50. Therefore, the correct answer is option (a) 50.