# Z-Score and Probability

1. An industrial sewing machine uses ball bearings that are targeted to have a diameter of 0.75 inch. The lower and upper specification limits under which ball bearings can operate are 0.74 inch and 0.76 inch, respectively. Past experience has indicated that the actual diameter of the ball bearings is approximately normally distributed, with a mean of 0.753 inch and a standard deviation of 0.004 inch. What is the probability that the ball bearing is

A: between the target and the actual mean?

B: between the lower specification limit and the target?

C: above the upper specification limit?

D: below the lower specification limit?

E: 93% of the diameters and greater than what value?

2. Part I. The fill amount of soft drink bottles is normally distributed, with a mean of 2.0 liters and a standard deviation of 0.05 liter. If bottles contain less than 95% of the listed net content (1.90 liters, in this case), the manufacturer may be subject to penalty by the state office of consumer affairs. Bottles that have a net content above 2.10 liters may cause excess spillage upon opening. What proportion of the bottles will contain

A: between 1.90 and 2.0 liters?

B: between 1.90 and 2.10 liters?

C: below 1.90 liters and above 2.10 liters?

D: 99% of the bottles contain at least how much soft drink?

E: 99% of the bottles contain an amount that is between which two values (symmetrically distributed) around the mean?

2. Part II. In an effort to reduce the number of bottles that contain less than 1.90 liters, the bottler in problem 6.30 sets the filling machine so that the mean is 2.02 liters. Under these circumstances, what are your answers in (a) through (e)?

A: between 1.90 and 2.0 liters?

B: between 1.90 and 2.10 liters?

C: below 1.90 liters and above 2.10 liters?

D: 99% of the bottles contain at least how much soft drink?

E: 99% of the bottles contain an amount that is between which two values (symmetrically distributed) around the mean?

3. Why are we able to substitute the normal distribution for the binomial probability distribution since the former is a continuous distribution and the latter is a discrete probability distribution?

4. It is known that one out of 3 people entering a large department store will make at least one purchase. If a random sample of 81 people is selected, what is the approximate probability that thirty or more of them will make at least one purchase? What is the probability that at most 40 of them will make at least one purchase?

5. An article in a sports magazine evaluated the relationship between physical fitness and stress. The research revealed that white-collar workers in good physical condition have only a 10% chance of developing a stress-related health problem. What is the probability that more than 60 in a random sample of 400 white-collar. Workers in good physical condition will develop stress-related illness?

6. DeKorte Telemarketing Inc. is considering purchasing a machine that randomly selects and automatically dials telephone numbers. It makes most of its calls during the evening, so calls to business phones are wasted. The manufacturer of the machine claims that their programming reduces the calling to business phones to 15 percent of all calls. To test this claim, the Director of Purchasing at DeKorte programmed the machine to select a sample of 150 phone numbers.

What is the probability that more than 30 of the phone selected are that of a business, assuming the manufacturer's claim is correct?

#### Solution Summary

The solution is comprised of detailed step-by-step calculation and explanation of the various problems related to Z-Score and Probability. This solution provides students with a clear perspective of the underlying mathematical aspects.