a random sample of 87 observations produced a mean=26.2 and a standard deviation s=2.5.find a95% confidence interval

Question

a random sample of 87 observations produced a mean=26.2 and a standard deviation s=2.5

Find a 95% interval

Find a 90% interval

Find a 99% interval

 

Answer 1

Hello, thank you for taking the time to post your question!

The underlying confidence interval formula will be the same in each of these scenarios, it’s just the critical value that changes based on the level of confidence that you are measuring.

The general equation here is

CI = x-bar +/- (Z* * s / sqrt(n)), so in each scenario we have

x-bar = mean = 26.2

s = standard deviation = 2.5

n = sample size = 87

the Z* changes based on the confidence level … for 95% you can use Z = 1.96, for 90% you can use Z = 1.645, and for 99% you can use z = 2.576

I hope that helps get you moving in the right direction! Feel free to reach out if you have any additional questions beyond that :)

Answer 2

A confidence interval is a range of values that is likely (but not guaranteed!) to contain the true population parameter that we are estimating. When calculating a confidence interval, we start with a point estimate (our 'best guess'), and then we add and subtract a margin of error to/from that point estimate. The point estimate is usually our sample statistic, and the margin of error is a function of the standard deviation, the sample size and our level of confidence (90%, 95%, etc.)

In this problem we are asked to calculate a confidence interval for a population mean, and we are told that the sample mean is 26.2, the sample standard deviation is 2.5 and the sample size is 87.

Our point estimate is the sample mean: 26.5

The margin of error is a t-critical value times the sample standard deviation divided by the square root of the sample size. t-crit*2.5/√87

We are using a t-critical value because we are given a sample standard deviation, rather than a population standard deviation. If we had been given a population standard deviation, we would use a z-critical value. To find t-critical values, we can refer to a table of critical values, or use technology.

For a sample size of 87 (87-1=86 degrees of freedom), the t-critical values are:

90% C.L. 1.663

95% C.L.: 1.990

99% C.L.: 2.634

(I used technology to find these critical values. If you use a table your values may be slightly different.)

Notice that as our level of confidence increases, our t-critical values increase. If we claim a higher level of confidence, we need to allow for a little extra 'wiggle room'- a wider margin of error.

90% Confidence interval: 26.2±1.663*2.5/√87 (25.75,26.65)

95% Confidence interval: 26.2±1.990*2.5/√87 (25.67, 26.73)

99% Confidence interval: 26.2±2.634*2.5/√87 (25.49,26.91)

Do you see that the interval becomes wider as our level of confidence increases?

In this explanation, I have attempted to provide you with the background needed to help you better understand confidence intervals, and we calculated the confidences manually. With technology, these intervals are very fast and easy to calculate.