# Normal distribution and Standard Error of the Mean

5. A professor at a local university noted that the grades of her students were normally distributed with a mean of 73 and a standard deviation of 11.

a. The professor has informed us that 7.93 percent of her students received grades of A. What is the minimum score needed to receive a grade of A?

b. Students who made 57.93 or lower on the exam failed the course. What percent of students failed the course?

c. If 69.5 percent of the students received grades of C or better, what is the minimum score of those who received C's?

6. The average weekly earnings of bus drivers in a city are $950 (that is, ) with a standard deviation of $45 (that is, ). Assume that we select a random sample of 81 bus drivers.

a. Compute the standard error of the mean.

b. What is the probability that the sample mean will be greater than $960?

c. If the population of bus drivers consisted of 400 drivers, what would be the standard error of the mean?

7. Ten percent of the items produced by a machine are defective. A random sample of 100 items is selected and checked for defects.

a. Determine the standard error of the proportion.

b. What is the probability that the sample will contain more than 2.5% defective units?

c. What is the probability that the sample will contain more than 13% defective units?

8. A random sample of 49 lunch customers was taken at a restaurant. The average amount of time the customers in the sample stayed in the restaurant was 33 minutes. From past experience, it is known that the standard deviation equals 10 minutes.

a. Compute the standard error of the mean.

b. What can be said about the sampling distribution for the average amount of time customers spent in the restaurant? Be sure to explain your answer.

c. With a .99 probability, what statement can be made about the size of the margin of error?

d. Construct a 99% confidence interval for the true average amount of time customers spent in the restaurant.

e. With a .99 probability, how large of a sample would have to be taken to provide a margin of error of 2.5 minutes or less?

9. A new brand of breakfast cereal is being market tested. One hundred boxes of the cereal were given to consumers to try. The consumers were asked whether they liked or disliked the cereal. You are given their responses below.

Response Frequency

Liked 60

Disliked 40

a. What is the point estimate of the proportion of people who will like the cereal?

b. Construct a 95% confidence interval for the true proportion of people who will like the cereal.

c. With a .95 probability, how large of a sample needs to be taken to provide a margin of error of 4% or less?

#### Solution Preview

5. a) Solution. Denote the grade of her student by X. By hypothesis, X follows normal distribution with a mean of 73 and a standard deviation of 11. Now, we need to find a such that

P(X >= a) = 7.93 %

By looking up the normal distribution table at http://people.hofstra.edu/faculty/Stefan_Waner/RealWorld/stats/normaldist.html, we have

P(X >= 88.52) = 7.93 %

So, the minimum score needed to receive a grade of A is 88.52.

b) Solution. By looking up the normal distribution table at http://people.hofstra.edu/faculty/Stefan_Waner/RealWorld/stats/normaldist.html, we have

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#### Solution Summary

The complete solution is included in an attached Word document.