# Probability calculation for normal and Poisson variables

(See attached file for full problem description)

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5.21 Assume that the number of network errors experience in a day on a local area network (LAN) is distributed as a Poisson random variable. The mean number of network errors experienced in a day is 2.4. What is the probability that in any given day

a.) Zero network errors will occur?

b.) Exactly one network error will occur?

c.) Two or more network errors will occur?

d.) Fewer that three network errors will occur?

6.1

a.) u = 0

sigma = 1

P (z<1.57)

x - u

Z = 1.57 - 0 = 1.57 = 1.57

1 1

P (z<1.57) = .9418

b.) P(z>1.84) = .0329

c.) P(z < 1.57<1.84) = .9418

d.) P(z < 1.57 or z >1.84) = .0253

6.9 The breaking strength of plastic bags used for packaging produce is normally distributed with a mean of 5 pounds per square inch and a standard deviation of 1.5 pounds per square inch. What proportions of the bags have a breaking strength of?

a) less than 3.17 pounds per square inch?

b) At least 3.6 pounds per square inch?

c) Between 5 and 5.5 pounds per square inch?

d) Between what two values symmetrically distributed around the mean will 95% of the breaking strengths fall?

6.11 A statistical analysis of 1,000 long distance phone calls made from the headquarters of the Bricks and Clicks Computer Corporation indicates that the length of these calls is normally distributed with u= 240 seconds and sigma = 40 seconds.

a) What is the probability that a call lasted less than 180 seconds?

b) What is the probability that a particular call lasted between 180 and 300 seconds?

c) What is the probability that a call lasted between 110 and 180 seconds?

d) What is the length of a particular call if only 1% of all calls are shorter?

6.27 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 the ball bearing 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 a 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 are greater than what value?

6.31 According to Investment Digest ('' Diversification and the Risk/Reward Relationship, " Winter 1994, 1-3), the mean of the annual return for common stocks from 1926 to 1992 was 12.4% and the standard deviation of the annual return was 20.6%. The article claims that the distribution of annual returns for common stocks is approximately bell-shaped and symmetric. Assume that the distribution is normally distributed with the mean and standard deviation given above. Find the probability that the return for common stocks will be.

a) Greater than 0%

b) Greater than10%

c) Greater than 20%

d) Less than -10%

7.9 The New York Times reported (Laurie J. Flynn, "Tax Surfing," The New York Times, March 25, 2002 C10) that the mean time to download the homepage from the Internal Revenue Service Web site ww.irs.gov was 0.8 seconds. Suppose that the download time was normally distributed with a standard deviation of 0.2 second. Is you select a random sample of 30 download times,

a) What is the probability that the sample mean is less than 0.75 second?

b) What is the probability that the sample mean is between 0.70 and 0.90 seconds?

c) The probability is 80% that the sample mean is between what two values symmetrically distributed around the population mean?

d) The probability is 90% that the sample mean is less than what value?

7.15 You plan to conduct a marketing experiment in which students are to taste one of two different brands of soft drink. Their task is to correctly identify the brand he or she tasted. You select a random sample of 200 students and assume that the students have no ability to distinguish between the two brands (each brand is equally likely to be selected.)

a.) What is the probability that the sample will have between 50% and 60% of the identifications correct?

b.) The probability is 90% that the sample percentage is contained within what symmetrical limits of the population percentage?

c.) What is the probability that the sample percentage of correct identifications is greater than 65%?

d.) Which is more likely to occur - more than 60% correct identifications in the sample of 200 or more than 55% correct identifications in a sample of 1,000? Explain.

7.31 Suppose that 5,000 sales invoices are separated into four strata. Stratum 1 contains 50 invoices, stratum 2 contains 500 invoices, stratum 1,000 invoices, and stratum 4 contains 3,450 invoices. A sample of 500 sales invoices is needed.

a) What type of sampling should you do? Why?

b) Explain how you would carry out the sampling according to the method stated in (a).

c) Why is the sampling in (a) not simple random sampling?

7.33 A simple random sample of n = 300 full-time employees is selected from a company list containing the names of all N = 5,000 full-time employees in order to evaluate job satisfaction.

a) Give an example of possible coverage error.

b) Give an example of possible non-response error.

c) Give an example of possible sampling error.

d) Give an example of possible measurement error.

7.50 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 the ball bearing can operate are 0.74 inch (lower) and 0.76 inch (upper). 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. If you select a random sample of 25 ball bearings, what is the probability that the sample mean is?

a) Between the target and the population mean of 0.753?

b) Between the lower specification limit and the target?

c) Above the upper specification limit?

d) Below the lower specification limit?

e) The probability is 93% that the sample mean diameter will be above what value?

8.63 The market research director for Dotty's department store wants to study women's spending on cosmetics. A survey is designed in order to estimate the proportion of women who purchase their cosmetics primarily from Dotty's department store, and the mean yearly amount that women spend on cosmetics. A previous survey found that the standard deviation of the amount women spend on cosmetics in a year is approximately $18.

a) What sample size is needed to have 99% confidence of estimating the population mean to within + $5?

b) What sample size is needed to have 90% confidence of estimating the population proportion to within + 0.045?

c) Based on the results in (a) and (b) how many of the store's credit cardholders should be sampled? Explain.

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The solution gives step by step procedure for the calculation of probability for normal and Poisson variable.