A simple random sample of 45 elements has been selected from a population of size 800. The sample mean is 500, and the sample standard deviation is 60. An approximate 95% confidence interval for the population mean is
A) 482.6 to 517.4
B) 440.0 to 560.0
C) 500.0 to 560.0
D) 440.0 to 500.0

Question

Quizplus · Accepted Answer

To calculate the confidence interval, we will use the formula:
CI = x̄ ± z* (σ/√n)
where x̄ is the sample mean, σ is the population standard deviation (which is unknown), n is the sample size, and z* is the critical value from the standard normal distribution corresponding to the desired confidence level (95% in this case).

Since we do not know the population standard deviation, we will use the sample standard deviation as an estimate. However, since the sample size is large enough (>30), we can assume that the sampling distribution of the mean is approximately normal and use z* instead of t* (which would be necessary for small sample sizes). The z* value for 95% confidence level is 1.96.

Substituting the values, we get:
CI = 500 ± 1.96 * (60/√45) = 500 ± 15.64

Therefore, the 95% confidence interval is (500-15.64) to (500+15.64), which simplifies to 482.36 to 517.64. Thus, the best choice is A.

A Simple Random Sample of 45 Elements Has Been Selected