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Chapter 8: Problem 17

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 obtained. 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

Expert verified
The probability that the sample mean for the 25 adult males being between 64 and 67 mm is found by calculating Z scores and looking up the probabilities under a standard normal distribution. The same approach is used for 100 adult males, applying the Central Limit Theorem to assume a normal distribution.

Step by step solution

01

Calculate Standard Error for sample of 25

Standard error (SE), which is the standard deviation of a sample's distribution, can be calculated using the formula SE = \(\frac{\sigma}{\sqrt{n}}\), where \(\sigma\) is the population's standard deviation (5 mm) and n is the sample size (25). Substituting these values, SE = \(\frac{5}{\sqrt{25}}\) = 1 mm.
02

Calculate Z scores and find probability for a sample size of 25

Z score (Z) is given by the formula \( Z = \frac{x - \mu}{SE} \), where x is the desired sample mean, \(\mu\) is the population mean (65 mm) and SE is the standard error (1 mm as calculated in Step 1). To find the probability of the sample mean being between 64 and 67 mm, we have to find the Z scores for these values and refer the resultant Z scores to the standard normal distribution table. Doing this, we have \( Z_1 = \frac{64 - 65}{1} = -1 \) and \( Z_2 = \frac{67 - 65}{1} = 2 \). Looking up these Z scores in the standard normal distribution table will give the required probabilities.
03

Calculate Standard Error for sample of 100

Still using the standard error formula from step 1, but with a new sample size of 100, we have SE = \(\frac{\sigma}{\sqrt{n}}\) = \(\frac{5}{\sqrt{100}}\) = 0.5 mm.
04

Calculate Z scores and find probability for a sample size of 100

Once again, calculate the Z scores by substituting the values into the Z score formula and lookup the scores in the standard normal distribution table. This time, the standard error calculated in step 3 is used. Similar to step 2, calculate the Z scores with x being 64 mm and 67 mm respectively. The results can then be looked up in the standard normal distribution table to get the probabilities.
05

Extra: Calculation for 'at least 68 mm'

For these parts of the question, it is noted that 'at least 68 mm' implies that the sample mean must be 68 mm or more, which means looking up the Z scores in the upper tail (right-hand side) of the standard normal distribution. The steps are similar except the probability retrieved from the standard normal table must be subtracted from one, as we are looking for the greater probability.

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