Elementary Statistics – Significance Testing of a Population Mean Examples

 

Example 1 (motivating example)

The Survey of Study Habits and Attitudes (SSHA) is a psychological test that measures the motivation, attitude toward school, and study habits of students. Scores range from 0 to 200. The scores for U.S. college students (ages 17 – 22) are normal with mean 115 and standard deviation 30. A teacher suspects that older students (at least 30 years old) have better attitudes toward school than younger students.

 

  1. Let  be the mean score of the older students. List the null and alternative hypotheses.

 

 

 

 

 

  1. Suppose the teacher suspected that older students simply have different attitudes toward school (not necessarily better attitudes). List the null and alternative hypotheses.

 

 

 

 

 

Suppose the teacher gives the test to an SRS of 50 college students who are at least 30 years old. Their mean score is 129.5. Furthermore, suppose that the population standard deviation for the scores of older students is 30.

 

We want to test the teacher’s original hypotheses.

 

  1. What is the probability of observing a sample mean of 129.5 or greater from a population with true mean 115?

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

  1. Suppose the teacher wanted to test at the significance level a = 0.01. Would the results be statistically significant?

 

 

 

Example 2

By law, an industrial plant can discharge no more than 500 gallons of wastewater per hour, on the average, into a neighboring lake. An environmental activist thinks that a certain plant is breaking the law.

 

She takes a random sample of 50 hours and finds the sample mean discharge to be 504.6 gallons. Suppose that we know that the standard deviation of wastewater discharge is 180.

 

Carry out the significance test (state the hypotheses, calculate the test statistic, calculate the p-value, and interpret the results).

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Example 3

For the general population of women between the ages 30 and 34, the mean diastolic blood pressure is 74.4 mm Hg. A researcher is interested in if diabetic women between the ages 30 and 34 have a different mean diastolic blood pressure than the general population.

 

He takes an SRS of 30 diabetic women between the ages 30 and 34 and determines their blood pressure. The sample mean diastolic blood pressure is 79.1 mm Hg (assume that the population standard deviation of the blood pressures of diabetic women is 9.8 mm Hg). He decides on a significance level of 0.01.

 

  1. Carry out the significance test (state the hypotheses, calculate the test statistic, calculate the p-value, and interpret the results).

 

 

 

 

 

 

 

 

 

 

  1. Create a 99% confidence interval for the unknown mean diastolic blood pressure. Does the confidence interval agree with your conclusion in part a?