The normal distribution plays a very important role in the science of statistical inference. You can determine the appropriateness of the normal approximation to an existing population of data by comparing the relative frequency distribution of a large sample of the data with the normal probability distribution.
Problem 4. 21 Assume that the length of time, x, between charges of a cellular phone is normally distributed with a mean of 10 hours and a standard deviation of 1.5 hours. Find the probability that the cell phone will last between 8 and 12 between charges.
Problem 4. 22 Suppose an automobile manufacturer introduces a new model that has an advertised mean in-city mileage of 27 miles per gallon. Although such advertisements seldom report any measure of variability, suppose you write the manufacturer for details on the tests and you find that the standard deviation is 3 miles per gallon. This information leads you to formulate a probability model for the random variable x, the in-city mileage for this car model. You believe that the probability distribution of x can be approximated by a normal distribution with a mean of 27 and a standard deviation of 3.
a.If you were to buy this model of automobile, what is the probability that you would purchase one that averages less than 20 miles per gallon for in-city driving? b.Suppose you purchase one of these new models and it does get less than 20 miles per gallon for in-city driving. Should you conclude that you probability model is incorrect?
Solution Part B. There are two possibilities that could exists: 1.The probability model is correct. You simply were unfortunate to have purchased one of the cars in the 1% that get less than 20 miles per gallon in the city. 2.The probability model is incorrect. Perhaps the assumption of a normal distribution is unwarranted, or the mean of 27 is an overestimate, or the stand deviation of 3 is an underestimate, or some combination of these errors occurred.
Keep in mind we have no way of knowing with certainty which possibility is correct, but the evidence points to the second on. We are again relying on the rare-event approach to statistical inference that we introduced earlier. In applying the rare-event approach, the calculated probability must be small (say, less than or equal to 0.05) in order to infer that the observed event is, indeed, unlikely.
Problem 4.22 Suppose that the scores x on a college entrance examination are normally distributed with a mean of 550 and the standard deviation of 100. A certain prestigious university will consider for admission only those applicants whose scores exceed the 90 th percentile of the distribution. Find the minimum score an applicant must achieve in order to receive consideration for admission to the university.
Solution Thus, the 90 th percentile of the test score distribution is 678. That is to say, an applicant must score at least 678 on the entrance exam to receive consideration for admission by the university.