# Probability: Binomial & Normal

Assume a binomial probability distribution with and . (Round all z values to 2 decimal places.)

Compute the following:

(a) The mean and standard deviation of the random variable. (Round your "σ" to 4 decimal places and mean to 1 decimal place.)

(b) The probability that X is 16 or more. (Use the rounded values found above. Round your answer to 4 decimal places.)

Probability

(c) The probability that X is 8 or less. (Use the rounded values found above. Round your answer to 4 decimal places.)

Probability

The closing price of Schnur Sporting Goods, Inc., common stock is uniformly distributed between $18 and 35 per share.Exercises

What is the probability that the stock price will be:

(a) More than $29? (Round your answer to 4 decimal places.)

Probability

(b) Less than or equal to $24? (Round your answer to 4 decimal places.)

Probability

The number of viewers of American Idol has a mean of 31 million with a standard deviation of 4 million. Assume this distribution follows a normal distribution.

What is the probability that next week's show will:

(a) Have between 33 and 39 million viewers? (Round z-score computation to 2 decimal places and your final answer to 4 decimal places. Round your answer to 4 decimal places.)

Probability

(b) Have at least 24 million viewers? (Round z-score computation to 2 decimal places and your final answer to 4 decimal places. Round your answer to 4 decimal places.)

Probability

(c) Exceed 43 million viewers? (Round z-score computation to 2 decimal places and your final answer to 4 decimal places. Round your answer to 4 decimal places.)

Probability

List the major characteristics of a normal probability distribution. (Select all that apply.)

Skewed.

Symmetrical.

Bell-shaped.

Asymptotic family of curves.

Uniform.

The manufacturer of a laser printer reports the mean number of pages a cartridge will print before it needs replacing is 12,275. The distribution of pages printed per cartridge closely follows the normal probability distribution and the standard deviation is 745 pages. The manufacturer wants to provide guidelines to potential customers as to how long they can expect a cartridge to last.

How many pages should the manufacturer advertise for each cartridge if it wants to be correct 99 percent of the time? (Round z value to 2 decimal places. Round your answer to the nearest whole number.)

Pages

A recent report in BusinessWeek indicated that 34 percent of all employees steal from their company each year. (Round z-score computation to 2 decimal places and your final answers to 4 decimal places.)

If a company employs 50 people, what is the probability that:

(a) fewer than 12 employees steal?

Probability

(b) more than 12 employees steal?

Probability

(c) exactly 12 employees steal?

Probability

(d) more than 12 but fewer than 22 employees steal?

Probability

Dotties Tax Service specializes in federal tax returns for professional clients, such as physicians, dentists, accountants, and lawyers. A recent audit by the IRS of the returns she prepared indicated that an error was made on 12 percent of the returns she prepared last year. Assuming this rate continues into this year and she prepares 68 returns. What is the probability that she makes errors on:

(a) More than 8 returns? (Round z-score computation to 2 decimal places and your final answer to 4 decimal places.)

Probability

(b) At least 8 returns? (Round z-score computation to 2 decimal places and your final answer to 4 decimal places.)

Probability

(c) Exactly 8 returns? (Round z-score computation to 2 decimal places and your final answer to 4 decimal places.)

Probability

A normal population has a mean of 20 and a standard deviation of 4.

(a) Compute the z value associated with 25 (Round your answer to 2 decimal places.)

Z =

(b) What proportion of the population is between 20 and 25? (Round z-score computation to 2 decimal places and your final answer to 4 decimal places. Round your answer to 4 decimal places.)

Proportion

(c) What proportion of the population is less than 18? (Round z-score computation to 2 decimal places and your final answer to 4 decimal places. Round your answer to 4 decimal places.)

Proportion

Customers experiencing technical difficulty with their Internet cable hookup may call an 800 number for technical support. It takes the technician between 150 seconds to 12 minutes to resolve the problem. The distribution of this support time follows the uniform distribution.

(a) What are the values for a and b in minutes? (Do not round your intermediate calculationsRound your answer to 1 decimal place.)

a

b

(b-1) What is the mean time to resolve the problem? (Do not round your intermediate calculations. Round your answer to 2 decimal places.)

Mean

(b-2) What is the standard deviation of the time? (Do not round your intermediate calculations. Round your answer to 2 decimal places.)

Standard deviation

(c) What percent of the problems take more than 5 minutes to resolve? (Do not round your intermediate calculations. Round your answer to 2 decimal places. Omit the "%" sign in your response.)

Percent %

(d) Suppose we wish to find the middle 50 percent of the problem-solving times. What are the end points of these two times? (Do not round your intermediate calculations. Round your answers to 3 decimal places.)

End point 1

End point 2

For the most recent year available the mean annual cost to attend a private university in the United States was $20,257. Assume the distribution of annual costs follows a normal probability distribution and the standard deviation is $4,325.

Ninety-nine percent of all students at private universities pay less than what amount? (Round z value to 2 decimal places and your final answer to the nearest whole number. Omit the "$" sign in your response.)

Amount $

A uniform distribution is defined over the interval from 2 to 5.

(a) What are the values for a and b?

a

b

(b) What is the mean of this uniform distribution? (Round your answer to 2 decimal places.)

Mean

(c) What is the standard deviation? (Round your answer to 4 decimal places.)

Standard deviation

(e) Find the probability of a value more than 2.6. (Round your answer to 2 decimal places.)

Probability

(f) Find the probability of a value between 2.9 and 3.7. (Round your answer to 4 decimal places.)

Probability

According to a government study among adults in the 25- to 34-year age group, the mean amount spent per year on reading and entertainment is $2,075. Assume that the distribution of the amounts spent follows the normal distribution with a standard deviation of $500. (Round z-score computation to 2 decimal places and your final answer to 2 decimal places. Omit the "%" sign in your response.)

(a) What percent of the adults spend more than $2,500 per year on reading and entertainment?

Percent %

(b) What percent spend between $2,500 and $3,050 per year on reading and entertainment?

Percent %

(c) What percent spend less than $1,275 per year on reading and entertainment?

Percent %

Among U.S. cities with a population of more than 250,000 the mean one-way commute to work is 24.3 minutes. The longest one-way travel time is New York City, where the mean time is 38.8 minutes. Assume the distribution of travel times in New York City follows the normal probability distribution and the standard deviation is 7.7 minutes.

(a) What percent of the New York City commutes are for less than 27 minutes? (Round your answers to 2 decimal places. Omit the "%" sign in your response.)

Percent %

(b) What percent are between 27 and 35 minutes? (Round your answers to 2 decimal places. Omit the "%" sign in your response.)

Percent %

(c) What percent are between 27 and 43 minutes? (Round your answers to 2 decimal places. Omit the "%" sign in your response.)

Percent

In establishing warranties on HDTV sets, the manufacturer wants to set the limits so that few will need repair at manufacturer expense. On the other hand, the warranty period must be long enough to make the purchase attractive to the buyer. For a new HDTV the mean number of months until repairs are needed is 36.84 with a standard deviation of 3.34 months.

Where should the warranty limits be set so that only 10 percent of the HDTVs need repairs at the manufacturers expense? (Round z-score computation to 2 decimal places. Round your final answers to 2 decimal places.)

warranty limits months

The monthly sales of mufflers in the Richmond, Virginia, area follow the normal distribution with a mean of 1,200 and a standard deviation of 225. The manufacturer would like to establish inventory levels such that there is only a 5 percent chance of running out of stock.

Where should the manufacturer set the inventory levels? (Enter the value without the separators. Round your answer to the nearest whole number.)

The mean of a normal probability distribution is 420; the standard deviation is 12.

(a) About 68 percent of the observations lie between what two values?

Value 1

Value 2

(b) About 95 percent of the observations lie between what two values?

Value 1

Value 2

(c) Practically all of the observations lie between what two values?

Value 1

Value 2

Assume that the mean hourly cost to operate a commercial airplane follows the normal distribution with a mean of $2,250 per hour and a standard deviation of $180.

What is the operating cost for the lowest 10 percent of the airplanes? (Round z value to 2 decimal places. Omit the "$" sign in your response.)

Operating cost $

See the attachment.

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The solution provides step by step method for the calculation of binomial and normal probabilities. Workings are given in the excel sheet.