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Chapter 7: Problem 52

The times that college students spend studying per week have a distribution that is skewed to the right with a mean of \(8.4\) hours and a standard deviation of \(2.7\) hours. Find the probability that the mean time spent studying per week for a random sample of 45 students would be a. between 8 and 9 hours b. less than 8 hours

Short Answer

Expert verified
a) The probability that the mean time spent studying per week for a sample of 45 students is between 8 and 9 hours is the area under the normal curve between Z1 and Z2. b) The probability that the mean time spent studying per week for a sample of 45 students is less than 8 hours is the area under the normal curve less than Z1.

Step by step solution

01

Parameters of Distribution

Identify the parameters of the distribution. Here, the population mean (碌) is 8.4 hours, the population standard deviation (蟽) is 2.7 hours and the sample size (n) is 45 students.
02

Calculate Standard Error

Calculate the standard error of the mean (SE), which is the standard deviation of the sampling distribution of the mean. The formula for SE is 蟽/鈭歯. So, SE = \(2.7 /\sqrt{45}\). Calculate the value of SE.
03

Calculate Z-scores for mean times

Calculate the Z-score for the given mean times. The Z-score is found using the formula Z = (X - 碌) / SE. a) For mean time between 8 and 9 hours, calculate two Z-scores, Z1 = (8-8.4) / SE and Z2 = (9-8.4) / SE. b) For mean time less than 8 hours, calculate the Z-score, Z1 = (8-8.4) / SE.
04

Find Probabilities

Use a standard normal distribution table to find probabilities corresponding to the calculated Z-scores. a) For mean time between 8 and 9 hours, find the area under the normal curve between Z1 and Z2, P(Z1 < Z < Z2). b) For mean time less than 8 hours, find the area under the normal curve less than Z1, P(Z < Z1).

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