{Exercise 9.37 (Algorithmic)} A study found that, in 2005, 12.5% of U.S. workers belonged to unions (The Wall Street Journal, January 21, 2006). Suppose a sample of 410 U.S. workers is collected in 2006 to determine whether union efforts to organize have increased union membership. Formulate the hypotheses that can be used to determine whether union membership increased in 2006. H0: p Ha: p If the sample results show that 50 of the workers belonged to unions, what is the sample proportion of workers belonging to unions (to 2 decimals)? Complete the following, assuming an level of .05. Compute the value of the test statistic (to 2 decimals). What is the p-value (to 4 decimals)? What is your conclusion?

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JeanaShupp JeanaShupp · Answer 1 · 2019-10-10T06:38:24+02:00

Let p represents the population proportion.

Then According to the given information, we have

[tex]H_0: p=0.125\\\\H_a: p>0.125[/tex]

∵ the alternative hypothesis is right-tailed , so the test is right-tailed test.

Given : For sample size of n=410 U.S. workers is collected in 2006 , 50 of the workers belonged to unions.

Then , sample proportion : [tex]\hat{p}=\dfrac{50}{410}\approx0.12[/tex] [Rounded to 2 decimals]

Test statistic: [tex]z=\dfrac{\hat{p}-p}{\sqrt{\dfrac{p(1-p)}{n}}}[/tex]

[tex]z=\dfrac{0.12-0.125}{\sqrt{\dfrac{0.125(1-0.125)}{410}}}\\\\=-0.306127891108\approx-0.31[/tex] [Rounded to 2 decimals]

The value of the test statistic : z= -0.31

P-value (Right -tailed test)=[tex]P(z>-0.31)=P(z<0.31)=0.6217195\approx0.6217[/tex] [Rounded to 4 decimals]

Since , the p-value (0.6217) is greater than the significance level (0.05), so we accept the null hypothesis.

Conclusion: We have sufficient evidence to reject the alternative hypothesis that the union membership increased in 2006.