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

Practice 1: Single Mean, Known Population Standard Deviation

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?

Means

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} } } {}:

Solution

• a: Ho:μ=2. 5Ho:μ=2. 5 size 12{H rSub { size 8{O} } :μ=2 "." 5} {} (or, Ho:μ2.5Ho:μ2.5 size 12{H rSub { size 8{O} } :μ <= 2 "." 5} {})
• b: H a : μ > 2 . 5 H a : μ > 2 . 5 size 12{H rSub { size 8{a} } :μ>2 "." 5} {}

Exercise 3

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

right-tailed

Exercise 4

What symbol represents the Random Variable for this test?

Solution

X ¯ X ¯ size 12{ {overline {X}} } {}

Exercise 5

In words, define the Random Variable for this test.

Solution

The mean time spent in jail for 26 first time convicted burglars

Exercise 6

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

Yes, 1.5

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={}} {}

• a. 3
• b. 1.5
• c. 1.8
• d. 26

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.

Solution

σ σ size 12{σ} {}

Exercise 9

State the distribution to use for the hypothesis test.

Solution

X ¯ ~ N ( 2 . 5 , 1.5 26 ) X ¯ ~ N ( 2 . 5 , 1.5 26 ) size 12{ {overline {X}} "~" N $$2 "." 5,1 "." 5 sqrt {"26"}$$ } {}

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.

Exercise 11

Find the p-value.

0.0446

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):

Solution

• a. Reject the null hypothesis

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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