Calculate the confidence interval for the expected value

Published on February 11, 2025

Statistics

The 95% confidence interval for the population mean is [66.04, 69.96], calculated using a sample mean of 68, a known variance of 22, and a sample size of 22. The minimum sample size needed to achieve a maximum confidence interval width of 1 is 338.

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Answer

We are given a normally distributed random variable

with unknown mean

and known variance

. We want to find a confidence interval for

at a confidence level of

.
A sample of size

is taken, and the sample mean is

First, we need to find the value of

such that for a standard normal distribution

.
For a 95% confidence level,

, and

. We need to find the z-score

such that

, or

.
From the standard normal distribution table or calculator,

. So,

The confidence interval for

is given by:

Here,

, and

.
Lower bound:

Upper bound:

The 95% confidence interval for

Now, we need to find the minimum sample size

such that the width of the confidence interval is at most

for the same confidence level

.
The width of the confidence interval is given by:

We want

Square both sides:

Since

must be an integer, we round up to the nearest integer:

Therefore, the value of

.
The confidence interval is

.
The minimum sample size

Final Answer: The final answer is

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Calculate the confidence interval for the expected value

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