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10.9 Hypothesis testing: two population means and two population

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This module provides a practice of Hypothesis Testing: Two Population Means and Two Population Proportions: as a part of Collaborative Statistics collection (col10522) by Barbara Illowsky and Susan Dean.

Student learning outcome

  • The student will conduct a hypothesis test of two means.

Given

The U.S. Center for Disease Control reports that the mean life expectancy for whites born in 1900 was 47.6 years and for nonwhites it was 33.0 years. (http://www.cdc.gov/nchs/data/dvs/nvsr53_06t12.pdf ) Suppose that you randomly survey death records for people born in 1900 in a certain county. Of the 124 whites, the mean life span was 45.3 years with a standard deviation of 12.7 years. Of the 82 nonwhites, the mean life span was 34.1 years with a standard deviation of 15.6 years. Conduct a hypothesis test to see if the mean life spans in the county were the same for whites and nonwhites.

Hypothesis testing: two means

Is this a test of means or proportions?

Means

State the null and alternative hypotheses.

  • H 0 : size 12{H rSub { size 8{0} } :} {}
  • H a : size 12{H rSub { size 8{a} } :} {}
  • H 0 : μ W = μ NW size 12{H rSub { size 8{O} } :μ rSub { size 8{W} } =μ rSub { size 8{ ital "NW"} } } {}
  • H a : μ W ≠ μ NW size 12{H rSub { size 8{a} } :μ rSub { size 8{W} }<>μ rSub { size 8{ ital "NW"} } } {}

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

two-tailed

What is the Random Variable of interest for this test?

X ¯ W − X ¯ NW size 12{ {overline {X}} rSub { size 8{W} } - {overline {X}} rSub { size 8{ ital "NW"} } } {}

In words, define the Random Variable of interest for this test.

The difference between the mean life spans of whites and nonwhites.

Which distribution (Normal or student's-t) would you use for this hypothesis test?

Explain why you chose the distribution you did for the above question.

Calculate the test statistic.

5.42

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

Find the p − size 12{p - {}} {} value:

0.0000

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 means are the same? Why or why not?

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Source:  OpenStax, Collaborative statistics using spreadsheets. OpenStax CNX. Jan 05, 2016 Download for free at http://legacy.cnx.org/content/col11521/1.23
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