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8.13 Practice 2: single mean, unknown population standard deviation

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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

Is this a test of means or proportions?

averages

State the null and alternative hypotheses.

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

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

two-tailed

What symbol represents the Random Variable for this test?

X ¯ size 12{ {overline {X}} } {}

In words, define the Random Variable for this test.

the mean time spent on death row for the 75 inmates

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

No

Calculate the following:

  • x ¯ = size 12{ {overline {x}} ={}} {}
  • 6 . 3 = size 12{6 "." 3={}} {}
  • n = size 12{n={}} {}

  • 17.4
  • s size 12{s} {}
  • 75

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

t − size 12{t - {}} {} test

State the distribution to use for the hypothesis test.

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

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

Find the p-value.

0.0015

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

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

  • Reject the null hypothesis

Discussion question

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

Questions & Answers

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Source:  OpenStax, Collaborative statistics (custom lecture version modified by t. short). OpenStax CNX. Jul 15, 2013 Download for free at http://cnx.org/content/col11543/1.1
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