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Practice 2: Single Mean, Unknown 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 unknown population standard deviation.

Given

A random survey of 75 death row inmates revealed that the mean length of time on death row is 17.4 years with a standard deviation of 6.3 years. Conduct a hypothesis test to determine if the population mean time on death row could likely be 15 years.

Hypothesis Testing: Single Mean

Exercise 1

Is this a test of means or proportions?

Solution

averages

Exercise 2

State the null and alternative hypotheses.

  • a. Ho:Ho: size 12{H rSub { size 8{O} } :} {}
  • b. Ha:Ha: size 12{H rSub { size 8{a} } :} {}

Solution

  • a. H o : μ = 15 H o : μ = 15 size 12{H rSub { size 8{O} } :μ="15"} {}
  • b. H a : μ 15 H a : μ 15 size 12{H rSub { size 8{a} } :μ <> "15"} {}

Exercise 3

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

Solution

two-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 on death row for the 75 inmates

Exercise 6

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

Solution

No

Exercise 7

Calculate the following:

  • a. x¯=x¯= size 12{ {overline {x}} ={}} {}
  • b. 6.3=6.3= size 12{6 "." 3={}} {}
  • c. n=n= size 12{n={}} {}

Solution

  • a. 17.4
  • b. ss size 12{s} {}
  • c. 75

Exercise 8

Which test should be used? In 1 -2 complete sentences, explain why.

Solution

tt size 12{t - {}} {}test

Exercise 9

State the distribution to use for the hypothesis test.

Solution

t 74 t 74 size 12{t rSub { size 8{"74"} } } {}

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.

Figure 1
Figure 1 (10.png)

Exercise 11

Find the p-value.

Solution

0.0015

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 Question

Does it appear that the mean time on death row could be 15 years? Why or why not?

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