Question

Suppose that the mean value of interpupillary distance (the distance between the pupils of the left and right eyes) for adult males is \(65 \mathrm{~mm}\) and that the population standard deviation is \(5 \mathrm{~mm}\). a. If the distribution of interpupillary distance is normal and a random sample of \(n=25\) adult males is to be selected, what is the probability that the sample mean distance \(\bar{x}\) for these 25 will be between 64 and \(67 \mathrm{~mm}\) ? At least \(68 \mathrm{~mm}\) ? b. Suppose that a random sample of 100 adult males is to be selected. Without assuming that interpupillary distance is normally distributed, what is the approximate probability that the sample mean distance will be between 64 and 67 \(\mathrm{mm} ?\) At least \(68 \mathrm{~mm} ?\)

Short answer

The question asks for probabilities. These are calculated using Z scores and a standard normal table. To calculate these, one must understand the use of Normal distribution and the Central Limit theorem. Also bear in mind that the larger the sample size, the closer the distribution gets to a normal distribution.

Step-by-step solution

Establish the given data

In this problem, the population mean \(\mu\) is 65mm, the population standard deviation \(\sigma\) is 5mm, and the sample size \(n\) is either 25 or 100.

Calculate the standard error and Z-score for part a

First, we calculate the standard error (SE) which is \(\sigma / \sqrt{n}\). Next, Z-score is calculated as \((mean - \mu) / SE\), for each boundary point - in this case, for 64mm and 67mm, and for \(at \ least \ 68mm\). The Z-score represents how many standard deviations our data are from the mean.

Find Probability for Part a - Sample Size of 25

The probabilities that the sample mean will be between 64mm and 67mm or at least 68mm are found by checking these z-scores on Z-table which gives us the probability under the normal curve up to that calculated z-score point.

Calculate the standard error and Z-score for Part b - Sample Size of 100

Repeat the procedure of Step 2 for the sample size of 100.

Find Probability for Part b - Sample Size of 100

Repeat the procedure of Step 3 for the new standard errors and calculated Z scores for the sample size of 100. Here we will use the Central Limit theorem to approximate the normal distribution because the distribution is not stated to be normal.

Interpret Results

The calculated probabilities represent the likelihood of the sample means respectively being between 64mm and 67mm or being at least 68mm. The larger the sample size, the more the distribution approaches a normal distribution (Central Limit Theorem).