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Practice 1: Single Mean, Known Population Standard Deviation

Module by: Susan Dean, Barbara Illowsky, Ph.D.. E-mail the authors

Summary: This module provides a practice of Hypothesis Testing of Single Mean and Single Proportion as a part of Collaborative Statistics collection (col10522) by Barbara Illowsky and Susan Dean.

Student Learning Outcomes

  • The student will conduct a hypothesis test of a single mean with known population standard deviation.

Given

Suppose that a recent article stated that the mean time spent in jail by a first–time convicted burglar is 2.5 years. A study was then done to see if the mean time has increased in the new century. A random sample of 26 first–time convicted burglars in a recent year was picked. The mean length of time in jail from the survey was 3 years with a standard deviation of 1.8 years. Suppose that it is somehow known that the population standard deviation is 1.5. Conduct a hypothesis test to determine if the mean length of jail time has increased. The distribution of the population is normal.

Hypothesis Testing: Single Mean

Exercise 1

Is this a test of means or proportions?

[ Show Solution ]

Exercise 2

State the null and alternative hypotheses.

  • a. HoHo size 12{H rSub { size 8{O} } } {}:
  • b. HaHa size 12{H rSub { size 8{a} } } {}:

[ Show Solution ]

Exercise 3

Is this a right-tailed, left-tailed, or two-tailed test? How do you know?

[ Show Solution ]

Exercise 4

What symbol represents the Random Variable for this test?

[ Show Solution ]

Exercise 5

In words, define the Random Variable for this test.

[ Show Solution ]

Exercise 6

Is the population standard deviation known and, if so, what is it?

[ Show Solution ]

Exercise 7

Calculate the following:

  • a. x¯=x¯= size 12{ {overline {x}} ={}} {}
  • b. σ=σ= size 12{σ={}} {}
  • c. sx=sx= size 12{s rSub { size 8{x} } ={}} {}
  • d. n=n= size 12{n={}} {}

[ Show Solution ]

Exercise 8

Since both σσ size 12{σ} {} and sxsx size 12{s rSub { size 8{x} } } {} are given, which should be used? In 1 -2 complete sentences, explain why.

[ Show Solution ]

Exercise 9

State the distribution to use for the hypothesis test.

[ Show Solution ]

Exercise 10

Sketch a graph of the situation. Label the horizontal axis. Mark the hypothesized mean and the sample mean x¯x¯ size 12{ {overline {x}} } {}. Shade the area corresponding to the p-value.

Blank horizontal axis of the sample mean.

Exercise 11

Find the p-value.

[ Show Solution ]

Exercise 12

At a pre-conceived α=0.05α=0.05 size 12{α=0 "." "05"} {}, what is your:

  • a. Decision:
  • b. Reason for the decision:
  • c. Conclusion (write out in a complete sentence):

[ Show Solution ]

Discussion Questions

Exercise 13

Does it appear that the mean jail time spent for first time convicted burglars has increased? Why or why not?

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