Question

Professor Smith has to grade assignments each week for her math class, and she suspects the week after her grading goes faster than usual. From previous experience, she knows the average amount of time it takes each week is 9 minutes per student and the standard deviation is 2.3 minutes per student. Assume a normal distribution. Last week, she had to grade assignments for 21 students. Let ?⎯⎯⎯⎯⎯ be the random variable representing the sample mean amount of time spent grading an assignment, for a sample of 21 students.

a. ?⎯⎯⎯⎯⎯ is normally distributed with a mean of _______________, and a standard error of the mean __________________. Round your answer to 2 decimal places.

b. Professor Smith finds that the week after midterms, her sample mean for 21 students is 7.44 minutes per student. Find the z-score associated to this sample mean, using the sampling distribution. Round your answer to two decimal places.

c. Find the probability that a randomly selected group of 21 students' assignments will take an average of 7.44 minutes or less to grade. Round your answer to 4 decimal places

d. Interpret the results. Is Professor Smith justified in concluding students' work is faster to grade the week after midterms?Select your answer from one of the following options.

• a.No, because probability is less than 0.05 that the average time per student was that low by chance
• b.No, because the probability is greater than 0.05 that the average time per student was that low by chance
• c.Yes, because probability is less than 0.05 that the average time per student was that low by chance
• d.Yes, because the probability is greater than 0.05 that the average time per student was that low by chance

Answer 1

Solution:
the average amount of time it takes each week is = µ = 9 minutes per student

the standard deviation is = σ = 2.3 minutes per student

Assume a normal distribution.

a.

n = 21 students

Let x̅ be the random variable representing the sample mean amount of time spent grading an assignment, for a sample of 21 students.

mean of distribution of sample means = µ = 9 minutes per student

standard error of the distrinution of sample mean = σ/√n = 2.3/√21 = 0.5019 minutes per student

x̅ is normally distributed with a mean of 9, and a standard error of the mean 0.50.

b.

n = 21 students

x̅ = 7.44 minutes per student

z score = (x̅-µ)/(σ/√n) = (7.44-9)/(2.3/√21) = -3.11

Answer : Z-score = -3.11

c.

Using Normal table

P( x̅ < 7.44 ) = P ( z < -3.11 ) = 0.0009

d. Interpret the results. Is Professor Smith justified in concluding students' work is faster to grade the week after midterms?Select your answer from one of the following options.

• c.Yes, because probability is less than 0.05 that the average time per student was that low by chance