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1. Has Support for Capital Punishment Changed Over Time? Surveys taken in [974 and in 2006 provide this data about whether or not US citizens favor the death penalt...

Question

1. Has Support for Capital Punishment Changed Over Time? Surveys taken in [974 and in 2006 provide this data about whether or not US citizens favor the death penalty:FavorOppose Total1974 2006937 19451410 2815870Find 95% confidence interva for the change the proportion of people supporting the death penalty between 1974 and 2006. Is it plausible that the proportion has not changed?Year

1. Has Support for Capital Punishment Changed Over Time? Surveys taken in [974 and in 2006 provide this data about whether or not US citizens favor the death penalty: Favor Oppose Total 1974 2006 937 1945 1410 2815 870 Find 95% confidence interva for the change the proportion of people supporting the death penalty between 1974 and 2006. Is it plausible that the proportion has not changed? Year



Answers

Statistics in the Media In May $2002,71 \%$ (0.71) of Americans favored the death penalty for a person convicted of murder. In May 2005,1005 adult Americans were asked by the Gallup Organization, "Are you in favor of the death penalty for a person convicted of murder?" Of the 1005 adults surveyed, 744 responded yes. The headline in the article reporting the survey results stated, "Americans' Views of Death Penalty More Positive This Year." Use a test of significance to support or refute this headline.

High in the abortion Data. We have a moment alone. nine which 5 million Americans favorite activity proportion and model so we take the US PS 3.65 population proportion. Number 1 10 105 Dominicans and 1: five. Are you in favor of death penalty? So this project is 100 years after his victory. And accordingly we find the and the proportion is coming out of the expo would end At 0.67 one. So highlight saying fewer americans a murder. That's the 90s. But we've always my hypothesis is turning out to be Equals to the .65 and already. But this is pure americans. So P is less than Your .65. There's already people that so will object whether we just support this uh segment we're Americans we will definitely be or not. We go with here are required the the point. So let's do it that small. The calculation give me half. So 7.0.63 71 negative b little point it's fine over There are 165 one negative points. It's fine or divide by N one durable by because we're done. Rita retirement was coming up to be well do -0.868 abuse technology began the value of P. That is coming up to be a little point 1928. Your point 192 weeks at the value of P which you are getting. So I made a graph like that. So distribution having had This is you know this is negative one living now negative zero point It's it's it it has statistics. So the area is covered 3.192. We try next area. So uh we have significant level is 0.05. Sophie's doesn't have to be greater than outcome. Therefore we pay to reject. Not only policy. Yeah. Okay Jack. Now we're going to start peel the coin. It's fine. We don't have an exception evidence to say that that we are americans. Uh We don't have sufficient evidence that the americans favor. Death penalties were both we refute the headline. Therefore we rip you. That's very nice. Thank you.

In this problem. The margin of the eater is given. E is equal to 3% which is equal to 0.3 level of significance. Al five given a 0.5 So the value of their Alpha B two can be calculated as that 0.5 by two, which is equal to their 0.0 25 which is equal to 1.96 The sample site can be calculated by the formula and is equal to P multiplication and one minus p. Multiplication. Jad Alfa by two by E Holy square So just putting the value I can write 20.641 minus 0.64 multiplication and 1.96 by 0.3 square. And on simplification, I get the value Age 984 So the n the at least sample side 984 people were surveyed. So this is the final answer for this problem. I hope you understand the solution

This question we are required to. Okay, we required my graph, which shows Americans that support. But they were not after conviction off Maeda and those who don't support it into a Sean Two models in the year. 1988 on the 1st 1 stating Explosive, It's true So Paige to X. Why two under on negative? It's bus wise because of 16 bits of its plus y cost to 16. This is a question this is too, so we can create a question. Three years question through. It's for us. Just took the other side. And then this becomes equation three. Now we can use substitution. We take our liquid that equation, Christian one. Then, if we do substitutes, this is what you get. Wherever there's why. Please put 16 plus X. That's what substitution in this guesses one about. Then this becomes 18. It's plus 16 by Wolf 16 times 12 One night two. Ah, blast golf. It's equals 2992 Then we added together like Tim's pull X plus 80 mics. 25 Indy five eggs is because too. Then 992 My last 192 It gives us 800 eight and because we're taking while 92 to the other side and they subjected for men into get 800 we make x the subject formula by dividing both sides by 25. Uh, then you get that too. On this side, you get that too. So that to a number of years, it's a yes. Now we also want to know which year. So you know the A. Yeah, people true way are in Pull out the years That's it. Take. And this will be 2020. This will be the 2020. Um Then we also want to find that the send age But he will be hope so we take from for the question three we get wise because 16 must it's which is 16 plus our exit stated to so that too last 16 equals to 48. So it before to eight and 48 to 4 to 8%. So listen sees for that for that year which is there 2020? What will be for dead? Been out in one presented you be against it This is there in which everything is equal 48% forded we're not in foot isn't against it

Following the solution to number 23 and this, uh, researchers asked the question, do you agree with the death penalty? And we're gonna find out how many are necessary, how many or the sample size is necessary in order to get a certain margin of error and that certain margin mayors within 3% points of the actual population proportion. So I wrote this formula up here in equals P. Hat times one minus P had. So that's the sample proportion times one minus the sample proportion times E star. That's the critical value divided by the margin of error quantity squared. And that's the formula is given in the book. But you can also derive that from the margin of error formula. So we're gonna use that to find the minimum sample size necessary in order to get Within in this case 3% points. So that's actually given in the problem. So the margin of air is .3, it's within 3% points of the actual population proportion. And 95% confidence means that the Z star is 1.96 Now, if you don't have that one memorized, you really should, that's probably the most common one. I always tell my students 1995 and 99 for sure. You need to have memorized in the 98 is also a good um Zis zis quarter to have memorized as well, but if you don't have to memorize, that's fine. You can use uh your table certainly works or you can use a T I 84 calculator, which is what I'm going to use to show you how I get that. So if I go to second distribution, second bars, it's you're gonna use this inverse norm down here, so option three, And then the area is going to be .95 and that represents the Middle 95%. The mean is always zero and the standard deviation is always one for these standardized scores. And then if you have the option to change the tail, it should be the center. Now if you don't have the option then you have to change it up a little bit, you would have to actually put the area as .025 or .975. Um and then It will understand it to be a left tail test if you have an older version, but hopefully you have a newer version, it makes things a little easier. And then when we paste, that's gonna give us these two critical values negative 1.96 and positive 1.96 after rounding and it really doesn't matter which one you use because you're gonna square it anyway. But typically you just use the the absolute value there. So it's one that's where I get that. 1.96. And the P. Hat was also given. So 64% say that they are in favor of the death penalty which means 36% say they are not in favor of the death penalty. So 360.64 is P. Hat and then one minus 10.64 point 36 And so from there we can actually just plug that into our calculators and our into our formula. And let's see what happens. So .64 times .36 times At 1.96 divided by .03 quantity squared. And that's gonna give you in equals 983 0.4. And then you always round up, you never round off. So early on whenever we start learning about rounding you would round this down to 983. And that's not what we do in statistics. You always round up not round off so that's going to be 984. Okay, so it's always gonna be a whole number whenever we're working with sample sizes, so 984 people are necessary to sample if we want to get within 3% points with 95° confidence and determining the proportion of people that are in favor of the death penalty.


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