Exam 6: Normal Probability Distributions

Solve the problem. -Suppose that replacement times for washing machines are normally distributed with a mean of 9.3 years and a standard deviation of 2 years. Find the replacement time that separates the top 18% from the bottom 82%.

Find the probability that in 200 tosses of a fair die, we will obtain at least 30 fives.

In a certain population, 11% of people are left-handed. Suppose that you plan to randomly select 100 people and ask each person whether they are left handed. Suppose that in calculating each of the probabilities below, you use the normal distribution as an approximation to the binomial but that you fail to use a continuity correction. In which cases will you obtain an answer that is too large? A: the probability that among the 100 people, at least 12 are left-handed B: the probability that among the 100 people, more than 12 are left-handed C: the probability that among the 100 people, between 10 and 14 inclusive are left-handed

Solve the problem. -A study of the amount of time it takes a mechanic to rebuild the transmission for a 1992 Chevrolet Cavalier shows that the mean is 8.4 hours and the standard deviation is 1.8 hours. If 40 mechanics are randomly selected, find the probability that their mean rebuild time exceeds 9.1 hours.

The Precision Scientific Instrument Company manufactures thermometers that are supposed to give readings of 0°C at the freezing point of water. Tests on a large sample of these thermometers reveal that at the freezing point of water, some give readings below 0°C (denoted by negative numbers) and some give readings above 0°C (denoted by positive numbers). Assume that the mean reading is 0°C and the standard deviation of the readings is 1.00°C. Also assume that the frequency distribution of errors closely resembles the normal distribution. A thermometer is randomly selected and tested. Find the temperature reading corresponding to the given information. -If 9% of the thermometers are rejected because they have readings that are too low, but all other thermometers are acceptable, find the temperature that separates the rejected thermometers from the others.

Provide an appropriate response. -Draw a normal distribution and identify the mean of x on the distribution. Discuss the symmetry and the total area under the curve. What is the probability that a value of x will be greater than the mean?

Solve the problem. -A bank's loan officer rates applicants for credit. The ratings are normally distributed with a mean of 200 and a standard deviation of 50. If 40 different applicants are randomly selected, find the probability that their mean is above 215.

Provide an appropriate response. -Lengths of pregnancies are normally distributed with a mean of 268 days and a standard deviation of 15 days. (a)Find the probability of a pregnancy lasting more than 250 days. (b)Find the probability of a pregnancy lasting more than 280 days. These two problems are solved almost exactly the same. Draw the diagram for each and discuss the part of the solution that would be different to finding the requested probabilities.

The Precision Scientific Instrument Company manufactures thermometers that are supposed to give readings of 0°C at the freezing point of water. Tests on a large sample of these thermometers reveal that at the freezing point of water, some give readings below 0°C (denoted by negative numbers) and some give readings above 0°C (denoted by positive numbers). Assume that the mean reading is 0°C and the standard deviation of the readings is 1.00°C. Also assume that the frequency distribution of errors closely resembles the normal distribution. A thermometer is randomly selected and tested. Find the temperature reading corresponding to the given information. -Find P40, the 40th percentile.

A math teacher gives two different tests to measure students' aptitude for math. Scores on the first test are normally distributed with a mean of 23 and a standard deviation of 5.1. Scores on the second test are normally distributed with a mean of 68 and a standard deviation of 10.4. Assume that the two tests use different scales to measure the same aptitude. If a student scores 28 on the first test, what would be his equivalent score on the second test? (That is, find the score that would put him in the same percentile.)

Solve the problem. -The weights of the fish in a certain lake are normally distributed with a mean of 12 lb and a standard deviation of 12. If 16 fish are randomly selected, what is the probability that the mean weight will be between 9.6 and 15.6 lb?

Find the probability that in 200 tosses of a fair die, we will obtain at exactly 30 fives.

The Precision Scientific Instrument Company manufactures thermometers that are supposed to give readings of 0°C at the freezing point of water. Tests on a large sample of these thermometers reveal that at the freezing point of water, some give readings below 0°C (denoted by negative numbers) and some give readings above 0°C (denoted by positive numbers). Assume that the mean reading is 0°C and the standard deviation of the readings is 1.00°C. Also assume that the frequency distribution of errors closely resembles the normal distribution. A thermometer is randomly selected and tested. Find the temperature reading corresponding to the given information. -A quality control analyst wants to examine thermometers that give readings in the bottom 7%. Find the reading that separates the bottom 7% from the others.

Provide an appropriate response. -Replacement times for T.V. sets are normally distributed with a mean of 8.2 years and a standard deviation of 1.1 years (based on data from "Getting Things Fixed," Consumers Reports). (a)Find the probability that a randomly selected T.V. will have a replacement time between 6.5 and 9.5 years. (b)Find the probability that a randomly selected T.V. will have a replacement time between 9.5 and 10.5 years. These two problems are solved almost exactly the same. Draw the diagram for each and discuss the part of the solution that would be different in finding the requested probabilities.

For the binomial distribution with the given values for n and p -n = 16 and p = .8

Provide an appropriate response. -Suppose you are asked to find the 20th percentile and the 80th percentile for a set of scores. These two problems are solved almost exactly the same. Draw the diagram for each and discuss the part of the solution that would be different in finding the requested probabilities.

A bank's loan officer rates applicants for credit. The ratings are normally distributed with a mean of 200 and a standard deviation of 50. If an applicant is randomly selected, find the probability of a rating that is between 170 and 220.

A coin is tossed 20 times. A person, who claims to have extrasensory perception, is asked to predict the outcome of each flip in advance. She predicts correctly on 16 tosses. What is the probability of being correct 16 or more times by guessing? Does this probability seem to verify her claim?

If Z is a standard normal variable, find the probability -Between 9.5 pounds and 11 pounds

A recent survey based on a random sample of n = 500 voters, predicted that the Independent candidate for the mayoral election will get 24% of the vote, but he actually gets 28%. Can it be concluded that the survey was done incorrectly?