Suppose that the mean value of interpupillary distance for all adult males is 65 mm, and the population standard deviation is 5 mm.

a) find the probability that a male selected at random will have an interpupillary distance less that 67.5 mm. assume that the distribution of interpupillary distances is approximately normal.

b) if a sample of 22 adult males is selected, what is the probability that the sample average distance x is between 64.3 and 66.4 mm?

c) given that the sample size in part (b) is small, explain why we could still use the z-table to answer the question.

a)

And we can find this probability using the z table or excel:

b)

c) And since the distribution of X is normal we can conclude that the distribution for the sample mean is also normal and for this reason we can use the z table no matter if the sample size is small.

Step-by-step explanation:

Previous concepts

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

The Z-score is "a numerical measurement used in statistics of a value's relationship to the mean (average) of a group of values, measured in terms of standard deviations from the mean".  

Part a

Let X the random variable that represent the interpupillary distance of a population, and for this case we know the distribution for X is given by:

Where and

We are interested on this probability

And the best way to solve this problem is using the normal standard distribution and the z score given by:

If we apply this formula to our probability we got this:

And we can find this probability using the z table or excel:

Part b

For this case the distirbution of the sample mean is also normal since the distribution for the random variable X is normal and is given by:

And for this case we want this probability:

And we can find this probability on this way:

And in order to find these probabilities we can find tables for the normal standard distribution, excel or a calculator.  

Part c

For this case since the sample mean is defined as:

If we find the expected value for this variable we got:

And for the variance we have:

And for this reason the deviation is

And since the distribution of X is normal we can conclude that the distribution for the sample mean is also normal and for this reason we can use the z table no matter if the sample size is small.

step-by-step explanation:

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step-by-step explanation:

if x = 2 you have to find x to finish the problem