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Normal distribution and Standard Error of the Mean

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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?

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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
...

Solution provided by:
Changping Wang, MA
Education
  • BSc , Wuhan Univ. China
  • MA, Shandong Univ.
Recent Feedback
  • "Your solution, looks excellent. I recognize things from previous chapters. I have seen the standard deviation formula you used to get 5.154. I do understand the Central Limit Theorem needs the sample size (n) to be greater than 30, we have 100. I do understand the sample mean(s) of the population will follow a normal distribution, and that CLT states the sample mean of population is the population (mean), we have 143.74. But when and WHY do we use the standard deviation formula where you got 5.154. WHEN & Why use standard deviation of the sample mean. I don't understand, why don't we simply use the "100" I understand that standard deviation is the square root of variance. I do understand that the variance is the square of the differences of each sample data value minus the mean. But somehow, why not use 100, why use standard deviation of sample mean? Please help explain."
  • "excellent work"
  • "Thank you so much for all of your help!!! I will be posting another assignment. Please let me know (once posted), if the credits I'm offering is enough or you ! Thanks again!"
  • "Thank you"
  • "Thank you very much for your valuable time and assistance!"
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