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Numerade Educator

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Problem 37 Medium Difficulty

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 400 U.S. workers is collected in 2006 to determine whether union efforts to organize have increased union membership.
a. Formulate the hypotheses that can be used to determine whether union membership increased in $2006 .$
b. If the sample results show that 52 of the workers belonged to unions, what is the $p$ -value for your hypothesis test?
c. At $\alpha=.05,$ what is your conclusion?

Answer

a. $H_{0} : p \leq 0.125, H_{a} : p>0.125$
b. $P=0.3821$
c. Fail to reject the null hypothesis $H_{0}$

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Video Transcript

We're told that in 2005 12.5% of American workers belong to unions. That gives us a proportion of 0.1 to 5. And we're also told that in 2006 we want to test to see if the proportion of American workers and unions has increased. We're told that in 2006 we have a sample of 400 workers for Part 80 were asked to formulate a hypothesis test for this scenario, so the alternative hypothesis would be that proportion is greater than 0.25 and therefore, the no hypothesis is that he is less than or equal to 0.1 to 5, and we can see that this is an upper tail a test. So that's our hypothesis. Test for B were as to calculate the P value for our scenario, and we're told that of the 400 workers in the sample, 52 were in unions, so that is a sample proportion, and that is equal to 0.13 So the next question ask ourselves, is how our sample proportions distributed and because end times p greater than or equal to five this could be verified easily. It's 400 times 0.1 to 5 and in times one minus P is also greater than or equal to five. Therefore, the sample proportions are approximately normally distributed, and so we're using the said statistic. So for sample proportions is that statistic can be estimated by and then plugging in the numbers, and that comes out to the 0.3. So now for the P value, so 0.3 would be somewhere around here. And so once we look when we look up 0.3 on the table, it's going to give us an area that corresponds to the cumulative probability, which is the area in the chart to the left of the said score. But RPI value is the area in the upper tail and from the said from the said table were given that this area is equal to 0.6179 and therefore the P value is equal to one minus 0.6179 So we have P value equals one minus 0.6179 in that equals 0.38 to 1. So that's the P value that that ends. Part B and Part C were asked what we would conclude if at an Alfa level of 0.5 So quite simply, a P value is bigger than Alfa. Therefore, we fail to reject the null hypothesis.