What is the 90% C.I.

Question

You intend to estimate a population mean � with the following sample.

50.5
60.3
58.8
62
67.4
64.7
69.2
52.6

You believe the population is normally distributed. Find the 90% confidence interval. Enter your answer as an open-interval (i.e., parentheses) accurate to two decimal places (because the sample data are reported accurate to one decimal place).

90% C.I. =rrec

Answer should be obtained without any preliminary rounding. However, the critical value may be rounded to 3 decimal places.

Answer 1

Let's start with the definition for a confidence interval. Based on the range that you set (i.e. 90%, 95%, 99%), it's the probability for the true value of the population parameter to be somewhere in that interval.

The formula for confidence interval calculation is:

(mean) ± z * ((sample standard deviation)/(√sample size))

To get the mean, you add up the values and divide them by the sample size. This gets you 60.69, after rounding to 2 decimal places.

The z score is your critical value, which is based on the confidence interval you wish you achieve. In this case, since the assignment is to calculate a 90% confidence interval, that would set your z-score to 1.645. You should look this up based on a given table, you don't necessarily have to remember this number, unless explicitly told to do so.

Your sample standard deviation is based on the difference between each value from the mean of all of your samples. The equation is:

SD = √{Sum[(X_i-Mean)²/(N - 1)]}. Basically, you take the difference between each individual value, square it, and add it to the rest of the values. Then divide it by the (sample size -1). Lastly, you take the square root of that value. In this case, your standard deviation is rounded to 6.64.

Lastly, you take the square root of the sample size.

Your final equation looks like this:

CI = 60.69 ± 1.65 * (6.64/(sqrt(8)) →

= 60.69 ± 3.86

The prompt tells you that you should format your answer in parentheses, therefore your answer would be

(56.83, 64.55).

Please ask about anything that you'd like explained better.