Statistical Hypothesis Testing
1. Which of the following is a valid statistical hypothesis?
(A) HA: p
2. In October 2010 Jimmy McMillan made national headlines while campaigning for Governor of New York on his party slogan “The Rent is Too Damn High.” Mr. McMillan only received 1% of the votes for Governor, but his campaign did create an interest in how much people pay for rent. Of particular interest is the proportion of all renters in the state of New York that pay $1400 or more per month in rent. To estimate the proportion of all renters in the state of New York that pay $1400 or more per month in rent, a 99% confidence interval will be calculated and the goal is that the margin of error will be no larger than .13. What is the minimum number of renters that would need to be selected to allow the calculation of a 99% confidence interval with margin of error no greater than .13? Please circle your final answer.
round up to 99
We would need a sample size of at least 99 renters.
3. A simple random sample of 100 renters in New York were selected and the amount paid in rent each month determined for each renter. 27 out of the 100 renters in the sample paid $1400 or more. If appropriate, use this information to calculate and interpret a 99% confidence interval for the proportion of all renters in the state of New York that pay $1400 or more per month in rent.
= (.1556, .3844)
We have 99% confidence that the proportion of all renters in the state of New York that pay $1400 or more per month in rent is between .1556 and .3844
4. Is the proportion of all renters in the state of New York that pay $1400 or more per month in rent in the confidence interval computed in question 3?
- The proportion of all renters in the state of New York that pay $1400 or more per month in rent is not known and hence we do not know if it is in the interval or not.
5. If all other things are held constant, what impact would increasing the sample size have on the margin of error and width of a confidence interval?
(B) Both the margin of error and the width would decrease.
6. Triggered by the bailouts of banks by the Bush and Obama administrations, the Tea Party movement is a populist, conservative/libertarian, grassroots political movement in the United States that grew throughout 2009 into a series of locally and nationally coordinated protests. Of interest is p = the proportion of all people who are active in the Tea Party movement that are under the age of 30. An unsubstantiated claim is that the proportion of all people who are active in the Tea Party movement under the age of 30 is .15, and of interest is to test this versus the alternative that the proportion of all people who are active in the Tea Party movement under the age of 30 is different from .15. State the appropriate null and alternative hypotheses that should be tested.
H0:
= .15 versus HA:
.15
7. Consider the information and hypotheses specified in question 6. A simple random sample of 120 Tea Party movement participants was selected, with 12 of these 120 people being under the age of 30. If appropriate, use this information to test the hypotheses stated in question 6 at the a = .10 level of significance.
We have a simple random sample,
Z =
= –1.53
p-value = 2P(Z
|–1.53|) = 2P(Z
1.53) = 2[1 – P(Z - (Calculator gives p-value = .1250)
a = .10 Since p-value > .10 we fail to reject H
There is insufficient evidence that the proportion of all people who are active in the Tea Party movement under the age of 30 is different from .15.8. Which of the following is the point estimate of the population proportion p?
(C)
9. America’s Got Talent made an appearance at Richmond’s Landmark Theater on Thursday, October 28. Suppose that 39.7% of all people who attended the show are age 75 or older. If a simple random sample of 90 attendees is selected and the proportion
who are age 75 or older determined, describe completely the sampling distribution of
We have a simple random sample, and since np = 90(.397) = 35.73 and n(1 – p) = 90(1 – .397) = 54.27 are both greater than 10, the sample size is large enough for the CLT to apply.
So
;
; and since the CLT applies the shape is normal. Hence
N(.397, .0516).
10. It is reported that 21% of all American adults weigh 225 pounds or more. At the America’s Got Talent show it was observed that many adults were fairly large, so it is of interest to test to determine if the proportion of all adults in attendance who weigh 225 pounds or more is greater than .21. State the appropriate null and alternative hypotheses that should be tested.
H0:
= .21 versus HA:
> .21
11. To test the hypotheses stated in question
10, a simple random sample of 40 adults who attended the America’s Got Talent show were selected. Of these 40 adults, 9 weighed 225 pounds or more. If appropriate, use this information to test the hypotheses stated in question 10 at the a = .10 level of significance.
The assumptions are not met –
= 40(.21) = 8.4, which is not
10, so we can not complete the test.