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Statistics / Fundamentals of Statistics 4 / Chapter 10.2 / Problem 30AYU

Fundamentals of Statistics | 4th Edition | ISBN: 9780321838704 | Authors: Michael Sullivan,III

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Solved: Small-Sample Hypothesis Test According to the U.S.

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Problem 30AYU

Problem

Small-Sample Hypothesis Test According to the U.S. Census Bureau, in 2009, 7.8% of Americans had a travel time to work of more than 60 minutes. An urban economist believes that the percentage of Americans who have a travel time to work of more than 60 minutes has increased since then. She randomly selects 80 working Americans and finds that 11 of them have a travel time to work that is more than 60 minutes. Test the urban economist’s belief at the α = 0.1 level of significance.

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QUESTION:

Problem 30AYU

Problem

Small-Sample Hypothesis Test According to the U.S. Census Bureau, in 2009, 7.8% of Americans had a travel time to work of more than 60 minutes. An urban economist believes that the percentage of Americans who have a travel time to work of more than 60 minutes has increased since then. She randomly selects 80 working Americans and finds that 11 of them have a travel time to work that is more than 60 minutes. Test the urban economist’s belief at the α = 0.1 level of significance.

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Answer:

Step 1 of 1

The sampling distribution of a count variable is only well-described by the binomial distribution is cases where the population size is significantly larger than the sample size. As a general rule, the binomial distribution should not be applied to observations from a simple random sample (SRS) unless the population size is at least 10 times larger than the sample size.

The Hypotheses can be expressed as

${H}_{0}:\ p\ =7.8\%$

${H}_{1}:\ p\ >7.8\%$

Or

${H}_{0}:\ p\ =0.078$

${H}_{1}:\ p\ >0.078$

at the α = 0.1 level of significance

Here, ${\widehat{p}}_{\ }=\frac{11}{80}=0.1375$

$n{p}_{\ }(1-{p}_{\ })$ = $80(0.078)(1-0.078)=5.7532$

$n{p}_{\ }(1-{p}_{\ })\ =5.7532\ <\ 10\$ and $n\leq{}0.05N$

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