21.4 Gun violence. The Harris Poll asked a sample of 1009 adults which causes of death they thought would become more common in the future. Gun violence topped the list. 706 members of the sample thought deaths from guns would increase. Although the samples in national polls are not true SRS’s they are close enough that our method gives approximately correct confidence intervals.
a) Say in words what the population proportion is for this poll: p is the proportion of adults from the population that Harris sampled. If the population is all adults in the U.S. then p is the proportion of all adults in the U.S. that think that deaths from guns will increase in the future.
b) Find the 95% confidence interval for p.
z*=1.96 for 95% confidence level
p^= 706/1009 = 0.6997 ~ 0.7
(margin of error – 2.8%)
(0.672,0.728)
c) Harris announced a margin of error of plus or minus 3%. How does this agree with b)? We got 2.8% approximately. Not bad.
21.5 Polling women. A New York Times poll on women’s issues interviewed 1025 women randomly selected from the U. S. excluding Alaska and Hawaii. Of the women in the sample, 482 said they do not get enough time to themselves. Although the samples in national polls are not true SRS’s they are close enough that our method gives approximately correct confidence intervals.
a) Explain in words what the parameter p is in this setting: p is the proportion of women in the US (excluding Alaska and Hawaii) that would have answered the question affirmatively.
b) Use the poll results to give the 95% confidence interval
z*=1.96 for 95% confidence level
p*=482/1025 ~ 0.47
(margin of error = 3.1%)
(0.439, 0.501)
c) Write a short explanation of your findings in b) for someone who knows no statistics: Are you kidding? If you were to repeat the poll many, many times then approximately 95% of the time the actual proportion would be in the confidence interval.
21.6 Gun Violence In Exercise 21.4, you gave a 95% confidence interval based on a random sample of n=1009. How large a sample would be needed to get a margin of error half as large in Exercise 21.4?
Answer: 4 times as large. The margin of error has a square root in it. You want it to be half as big so you need a 4 under the square root to get a 2 out side the square root. Try it and see!!
21.7 The effect of sample size. An opinion poll finds that 60% of its sample prefer balancing the federal budget to cutting taxes. Give a 95% confidence interval for the proportion of all adults who feel this way assuming the result of p^ - 0.6 comes from sample size
a) n = 750
z*=1.96 for 95% confidence level
p*=0.6
(margin of error 3.5%)
(0.565, 0.635)
b) n = 1500
z*=1.96 for 95% confidence level
p*=0.6
(margin of error 2.5%)
(0.575, 0.625)
c) n = 3000
z*=1.96 for 95% confidence level
p*=0.6
(margin of error 1.75%)
(0.5825, 0.6125
Note that this margin of error is ˝ of the original and the size of n is 4 times as big.
d) Explain briefly what your results show about the effect of increasing the size of a sample: The larger the sample the better your result should be and so the margin of error and the confidence interval should get smaller.
21.8 Random Digits. We know that the proportion of 0’s among a large set of random digits is p=0.1 because all 10 possible digits are equally probable. The entries in a table of random digits are a sample from the population of all random digits. To get an SRS of 200 random digits look a the first digit of the 200 five-digit groups in lines 101 to 125 of Table A in the back of the book. How many 200 digits are 0s? Give a 95% confidence interval for the proportion of 0s in the population from which these are a sample. Does your interval cover the true parameter value, p = 0.1?
I used Excel to generate the results a few times. Out of 200 I got 19. 18. 21. 26. 21. 25 zeros in 6 of the tries. So for example suppose we use p^=18. Note that n = 200
b) Find the 95% confidence interval for p.
z*=1.96 for 95% confidence level
p^= 18/200 = 0.09
(margin of error – 4.0%)
(0.05,0.13)
Since our actual p is 0.1. It is in the interval.
However if I look at n=26
p^=26/200 = 0.13
z*=1.96 for 95% confidence level
p^= 18/200 = 0.09
(margin of error – 4.0%)
(0.05,0.13)
Since our actual p is 0.1. It is barely outside interval.
So this means that it is one of the 5% of the intervals that does not contain p.
21.9 Tossing a thumbtack. If you toss a thumbtack on a hard surface, what is the probability that it will land point up? Estimate this probability p by tossing a thumbtack 100 times.
I couldn’t find a real thumbtack but found something like a thumbtack. I had to cut a hook off of it but then I tossed it 100 times and got 56 times. So: p^ = 56/100 = 0.56.
The 100 tosses are an SRS of size 100 out of the population of all tosses. The proportion of these 100 tosses that land point up is the sample proportion p^. Use the result of your tosses to give a 95% confidence interval for p.
z*=1.96 for 95% confidence level
p^= 0.56
(margin of error – 9.7%)
(0.463,0.657)
Write a brief explanation of your findings for someone who knows no statistics. You have to be kidding!! If you were to do this experiment many, many times and found the confidence intervals for each, then 95% of the time the actual proportion of the time that the thumbtack would land point up would be in the confidence interval. Roughly speaking, the probability is 0.95 that the actual proportion is in the confidence interval.
21.10. Don’t forget the basics. Heterosexuals were surveyed to find how many had multiple sexual partners.
There might to be a bias built in to this survey because, it seems to me, that people will lie about their sex lives.
21.11 Count Buffon’s coin. The 18th century French naturalist Count Buffon tossed a coin 4040 times. He got 2048 heads.
Give a 95% confidence interval for the probability that Buffon’s coin lands heads up.
p^ = 2048/4040 = 0.57
z*=1.96 for 95% confidence level
(margin of error = 1.5%)
(0.555, .585)
How confident are you that the probability is not ˝.
I guess I am 95% confident. Since this is the 95% confidence interval and it doesn’t contain 0.5, it means to me that 95% of the time my interval would contain the actual probability.