91Ó°ÊÓ

Given a standard normal distribution. find the normal curve area under the curve which lies (a) to the left of \(z=1.43\); (b) to the right of \(z=-0.89\) : (c) between \(z=-2.16\) and \(z=-0.65\) (el) to the left of; \(=-1.39\); (e) to the: right of \(z=1.96\) : (T) between \(z=-0.48\) and \(z=1.74\).

Find the value of \(z\) if the area under a standard (a) to the right of \(z\) is 0.3622 ; (b) to the left of \(z\) is \(0.1131 ;\) (c) between 0 and \(z\), with \(z>0,\) is 0.4838 ; (d) between \(-z\) and \(z\), with \(z>0\), is 0.9500 .

A soft-drink machine is regulated so that it discharges an average of 200 milliliters per cup. If the amount of drink is normally distributed with a standard deviation equal to 15 milliliters, (a) what fraction of the cups will contain more than 224 milliliters? (b) what is the probability that a cup contains between 191 and 209 milliliters? (c) how many cups will probably overflow if 230 milliliter cups are used for the next 1000 drinks? (d) below what value do we get the smallest \(25 \%\) of the drinks?

The finished inside diameter of a piston ring is normally distributed with a mean of 10 centimeters and a standard deviation of 0.03 centimeter. (a) What proportion of rings will have inside diameters exceeding 10.075 centimeters? (b) What is the probability that a piston ring will have an inside diameter between 9.97 and 10.03 centimeters? (c) Below what value of inside diameter will \(15 \%\) of the piston rings fall?

A lawyer commutes daily from his suburban home to his midtown office. The average time for a one-way trip is 24 minutes, with a standard deviation of 3.8 minutes. Assume the distribution of trip times to be normally distributed. (a) What is the probability that a trip will take at least \(1 / 2\) hour? (b) If the office opens at 9: 00 A.M. and he leaves his house at 8: 45 A.M. daily, what percentage of the time is he late for work? (c) If he leaves the house at 8: 35 A.M. and coffee is served at the office from 8:50 A.M. until 9:00 A.M., what is the probability that he misses coffee? (d) Find the length of time above which we find the slowest \(15 \%\) of the trips. (e) Find the probability that 2 of the next 3 trips will take at least \(1 / 2\) hour.

In the November 1990 issue of Chemical Engineering Progress, a study discussed the percent purity of oxygen from a certain supplier. Assume that the mean was 99.61 with a standard deviation of 0.08 . Assume that the distribution of percent purity was approximately normal. (a) What percentage of the purity values would you expect to be between 99.5 and \(99.7 ?\) (b) What purity value would you expect to exceed exactly \(5 \%\) of the population?

The average life of a certain type of small motor is 10 years with a standard deviation of 2 years. The manufacturer replaces free all motors that fail while under guarantee. If he is willing to replace only \(3 \%\) of the motors that fail, how long a guarantee should he offer? Assume that the lifetime of a motor follows a normal distribution.

The heights of 1000 students are normally distributed with a mean of 174.5 centimeters and a standard deviation of 6.9 centimeters. Assuming that the heights are recorded to the nearest half-centimeter, how many of these students would you expect to have heights (a) less than 160.0 centimeters? (b) between 171.5 and 182.0 centimeters inclusive? (c) equal to 175.0 centimeters? (d) greater than or equal to 188.0 centimeters?

A company pays its employees an average wage of \(\$ 15.90\) an hour with a standard deviation of \(\$ 1.50\). If the wages are approximately normally distributed and paid to the nearest cent, (a) what percentage of the workers receive wages between \(\mathrm{S} 13.75\) and \(\mathrm{S} 16.22\) an hour inclusive? (b) the highest \(5 \%\) of the employee hourly wages is greater than what; amount?

The weights of a large number of miniature poodles are approximately normally distributed with a mean of 8 kilograms and a standard deviation of 0.9 kilogram. If measurements arc recorded to the nearest tenth of a kilogram, find the fraction of these poodles with weights (a) over 9.5 kilograms: (b) at most 8.6 kilograms; (c) between 7.3 and 9.1 kilograms inclusive.