Question

A researcher finds a mean of 80, and a standard deviation of 6, based on a sample of 100 observations. What is the 95% confidence interval?

1) 68.1, 91.9
2) 75.2, 84.8
3) 78.8, 81.2
4) 79.5, 80.5

Answer 1

The 95% confidence interval for a sample mean of 80 with a standard deviation of 6 and a sample size of 100 is (78.8, 81.2) after rounding to one decimal place.

Step-by-step explanation:

To calculate the 95% confidence interval for a sample mean when the population standard deviation is known, we can use the formula for a z-interval:

X ± Z*(σ/√n)

Where X is the sample mean, Z* is the z-score corresponding to the desired confidence level, σ is the population standard deviation, and n is the sample size. For a 95% confidence interval and a sample size of 100, the z-score (Z*) is approximately 1.96.

Step-by-step calculation:

1. Identify the sample mean (X) which is 80.
2. Identify the sample standard deviation (σ), which is 6.
3. Identify the sample size (n), which is 100.
4. Calculate the standard error (σ/√n) = 6 / √100 = 6 / 10 = 0.6.
5. Calculate the margin of error using the z-score: 1.96 * 0.6 = 1.176.
6. Calculate the confidence interval: 80 ± 1.176 = (78.824, 81.176)

After rounding to one decimal place, the confidence interval is (78.8, 81.2).