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Textbook by: Kathy Chu, Ph.D.. E-mail the author

# Practice 2: Single Mean, Unknown 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 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?

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?

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?

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.

### Exercise 11

Find the p-value.

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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