Question

A study of 65 bolts of carpet showed that their average length was 78.2 yards. The standard deviation of the population is 1.6 yards. Which of the following is the 80% confidence interval for the mean length per bolt of carpet?

Answer 1

Imagine a normal distribution in your head. A normal distribution contains 100% of scores. With the largest percentage of scores clustering towards the mean (or centre).

A confidence interval is just an interval estimate range in which the researcher would expect to find the mean of the general population, a certain percentage of the time (in your case 80% of the time) IF the sample population is indeed a part of the general population.

So If you have a confidence interval of 80%. Subtract 80% from the 100% of scores on the distribution. This would leave you with 20%. This 20% is the percentage of the distribution that if the population mean happens to falls within. Would mean that the sample population is NOT a part of the normal distribution.

On a normal distribution, the remaining 20% of the distribution must be divided up between the two tails. That means 10%, of the 20% must be in each tail of the distribution. So if 10% of the scores is in each end of the distribution. Look at your Z-tables (you should have this in your text book). Look up the correlating Z-score for a distribution with 10% of scores in the tail.

The answer will be 1.29.

We know that the low end (or left) of the distribution (with 10% in the tail) are all negative numbers. Therefore the 1.29 will be -1.29. And the higher end (right) of the distribution will be 1.29.

Since you are looking at a sample population. You need to use standard error (NOT standard deviation. Standard deviation is only used on general populations NOT samples). In order to calculate the standard error. Use this formula:

Standard deviation divided by the square root of the sample size.

1.6/ square root of 65

square root of 65= 8.0622577
1.6 / 8.0622577= 0.19

Your standard Error is 0.19.

Now in order to turn the upper and lower z-scores that you just found into raw numbers. Use this formula:

M Lower= -Z(Standard Error)+ Sample Mean
M Upper= Z(Standard Error) + Sample Mean


When we plug the numbers into the formula we get:

M Lower= -1.29 (.19) + 78.2 = 77.9549 (we rounded up to the second decimal)
M Upper= 1.29 (.19) + 78.2 = 78.4451


So our lower confidence interval range limit is = 77.9549
Our higher confidence interval range limit is = 78.4451



You can check this answer by subtracting the lower and upper limits from the mean of the sample. If each number is exactly the same distance away from the mean, then the range limits are correct.

Ex= 77.9549 - 78.2= -0.2451
78.4451 - 78.2 = 0.2451