Question

Two different samples will be taken from the same population of test scores where the population mean and standard deviation are unknown. The first sample will have 25 data values, and the second sample will have 64 data values. A 95% confidence interval will be constructed for each sample to estimate the population mean. Which confidence interval would you expect to have greater precision (a smaller width) for estimating the population mean?

Answer 1

Confidence Interval with sample size 25 = Broader, less precision; Confidence Interval with sample size 64 = Narrower, more precision

Explanation:

Confidence Interval is the range around sample statistic, which contains the actual population parameter. Confidence level is the percentage probability, with which the population parameter is expected to be in confidence interval.

When sample size increases : the margin of error, ie sampling error between population parameter & sample statistic decreases. The reduced margin of error increases the level of confidence & precision in confidence interval range. So, the confidence interval range becomes narrower.

Hence, confidence interval becomes narrower & has more precision, when sample size increases from sample number = 25 to sample number = 64.

Answer 2

The sample consisting of 64 data values would give a greater precision.

Explanation:

The width of a (1 - α)% confidence interval for population mean μ is:

`\text{Width}=2\cdot z_(\alpha/2)\cdot (\sigma)/(√(n))`

So, from the formula of the width of the interval it is clear that the width is inversely proportion to the sample size (n).

That is, as the sample size increases the interval width would decrease and as the sample size decreases the interval width would increase.

Here it is provided that two different samples will be taken from the same population of test scores and a 95% confidence interval will be constructed for each sample to estimate the population mean.

The two sample sizes are:

n₁ = 25

n₂ = 64

The 95% confidence interval constructed using the sample of 64 values will have a smaller width than the the one constructed using the sample of 25 values.

Width for n = 25:

`\text{Width}=2\cdot z_(\alpha/2)\cdot (\sigma)/(√(25))=(1)/(5)\cdot [2\cdot z_(\alpha/2)\cdot \sigma]`

Width for n = 64:

`\text{Width}=2\cdot z_(\alpha/2)\cdot (\sigma)/(√(64))=(1)/(8)\cdot [2\cdot z_(\alpha/2)\cdot \sigma]`

Thus, the sample consisting of 64 data values would give a greater precision