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Assume that the cakculated test sratistic 1,28 (use this value even if you found another value vour cakulations}. Then; what is the conclusion the hypothesis test? ...

Question

Assume that the cakculated test sratistic 1,28 (use this value even if you found another value vour cakulations}. Then; what is the conclusion the hypothesis test? Use the critical t-value from QuestionFail t0 Rcject Ho becxarsc thc ckulated tcst stanstc Icss than tc crincal tvalucFail to Rcjcct Ho bcrausc thc cakculatcd tcst statisticthzn thc criticalt-valucRejcct Ho because Uc cakulatcd test stansnc is less than the crineal EvJlue-Noncortcrs Rcicl Ho bccause Wc clkuljicd [cst suansnc I5 mcrc [

Assume that the cakculated test sratistic 1,28 (use this value even if you found another value vour cakulations}. Then; what is the conclusion the hypothesis test? Use the critical t-value from Question Fail t0 Rcject Ho becxarsc thc ckulated tcst stanstc Icss than tc crincal tvaluc Fail to Rcjcct Ho bcrausc thc cakculatcd tcst statistic thzn thc criticalt-valuc Rejcct Ho because Uc cakulatcd test stansnc is less than the crineal EvJlue- Nonc ortcrs Rcicl Ho bccause Wc clkuljicd [cst suansnc I5 mcrc [han (hc crincalt-valuc



Answers

Consider the following hypothesis test:
$$\begin{array}{l}{H_{0} : \mu \geq 20} \\ {H_{\mathrm{a}} : \mu<20}\end{array}$$
A sample of 50 provided a sample mean of $19.4 .$ The population standard deviation is 2 .
a. Compute the value of the test static.
b. What is the $p$ -value?
c. Using $\alpha=.05,$ what is your conclusion?
d. What is the rejection nle using the critical value? What is your conclusion?

The following is a solution video to number five. And we're doing a hypothesis test with a difference of two population proportions. Uh One where there is no difference between the two proportions and then one where P one minus P two is greater than zero or P one is greater than P two. And we're testing at the 10% level of significance and we're giving this data here. So the first sample sizes and the second sample size for 1200. And the P one hat was 10.42 and the P two is .4. Okay, so um you can do this on some of these if you just if we can get the number of favorable and then we can use a calculator and it makes it a little bit nicer. So the the way that we get the number of favorable, we just take in one times P one hat. So if we take 1200 times 12000.42, you get five oh four. And then for X to do the same thing 1200 times four and 480. So with that information we can actually uh use the calculator. So five or four and 4 80 if you go to stat and then tests and then we go to this to prop Z test And we go 504 And then that's had a 1200 And then 4 80. And that's out of 1200 and then P one is great MP two. So that's all set up and ready to go uh And we have a z value about one And AP value of about .16. Now we're going to do this the critical region method because that's the way that they asked us to do but we at least have a decent idea of what's going on here. So the Z value is 0996 and we're going to do the critical region method. Okay, so to do that, I'm going to go back to my calculator. It's a 10% level of significance. If I go to second distribution and then I go to inverse norm And then I type in .10 for my area and this is a greater than so it's a right till test. My critical value is 1.282. So 1.282 is my critical region. Okay. And anything greater than that? I reject. Well, my test statistic lens in the non rejection regions I want to say fail to reject. Okay. And then we're also asked to uh find the p value and the p value we already found on the calculator if you don't remember it. P value was about 0.16. Right? So that would tell you to fail to reject as well because it's greater than 0.10. Okay, part two I don't believe we're as lucky on this one where we can just multiply because we get decimals and we can't use decimals here. So um let's go ahead and just find the test statistic using the formula. So it's the P one hat minus P two hat which would be 20.61 minus point 0.67 And then minus the hypothesized value which is zero divided by square root. Big screwed of 6 1 p one hat times one minus P one hat which is 10.39. And that's divided by the sample size 5 50 and then plus 0.67 Times 1 -6 7 or 3 3 Divided by 600. And that's going to give you if you plug that in a calculator carefully you should give you negative to 12 as a test statistic. Now the Z value we look at alpha equals five and it's a two tailed tests. So the Z star the critical value is do this a couple ways. Um No sorry second distribution so we're gonna go to inverse norm and what I like to do is five. If I do one minus that It's .95. And then for two tailed tests I just put that rascal in the center there and that's gonna give me my to test my two critical values, positive negative 1.96. So that's what I can write down. So z star or the critical value is positive, negative 1.96. Okay, So anything to the left of negative 1.96 and anything to the right of positive 1.96. We would reject and this negative 2.12 lands in the rejection region. So I'm going to reject. H. Not. Okay, so the p value for two tailed tests are a little bit trickier. Uh The p value is going to be the probability that we get a test statistic less than negative 2.12 or greater than positive 2.12. So I need to basically just find the probability that is less than negative 2.12 and then multiplied by two. So my lower bound is negative infinity. My profound. Is that negative 2.12? And then you zero standard deviations one And then I paste. And then I get that. Now I need to multiply that by two to account for the other tail. So it's .034. As my P values, the p value is 0.034.

So we're given that an is equal to 300. Alfa is equal to 0.5 Mm yeah, which is our level. Significance on X is equal to 27. Therefore we can kill. Okay, X over n which is equal to 27. Okay, 300 literally. Call. Does your point. Zero? No. So, Donald, Hypothesis H nine is a P is equal to 01 andare alternative hypothesis is a P is less thens you're going so we can now use our formula to find those eat statistic. Yes. Yeah. Square P time zone minus p over. And which is he called? 009 minus zero. Mhm. Over the square was your one Times one nice. You should you. And over 300. She'll give us natives Europe 58 Now we can find g critical value, which is equal to negative 1.645 She was in our value. If she opens your Fife. Andi since r p value a z less than two negatives Europe on the side eight, which is equal to 0.280 man. Okay, it's there for the p value is greater than yeah, pretty cool significance level and therefore you can fail to reject the age, not there is not enough evidence to conclude that the proportion of test, which are wrong, is less than 10%.

Even the following information were asked to find a P value, associate it with the given information here and compare our P value to an Alfa 0.5 to test these hypotheses. So the first thing we need to find is a test statistic. And because we're working with proportions as given by this P bar here, we are going to compute a pool variance. So the formula for our P bar is the first sample size times the first sample proportion plus the second sample size times a second sample proportion over some of the first and second sample sizes. So in this situation that is equal to 200 times 0.22 plus 300 times 0.16 divided by 300 plus 200 and we get a test statistic of 0.184 So this is our P bar and now we can use this this value to compute a Z score. So our Z score is equal to the difference of our proportions. Divided by the square root of are pulled variants times one minus our pool variance times one over our first sample size plus one over our second sample size and all of that under a square root. So this is equal to 0.22 minus 0.16 over the square root of our P bar. We just found out to be 0.184 times one minus 10.184 one minus 10.184 times. Um, one over 200 plus one over 300 and all of that square rooted. And we get a Z test statistic of approximately 1.70 so we'll draw our normal curve. Z equals zero lies in the middle. Z equals 1.7 lives here and now we can check our oh, so part, eh? Heart lead. Um, so important. We were asked to find a p value, but we need to find the direction of, um of the hypothesis in order to discover our p value. So since this difference of means is greater than zero sorry, the difference of proportions is greater than zero. We're going to be doing a right tail t test Z test. Sorry. So we're looking at this value here, and this value represents the probability that Z is greater than or equal to 1.7, and that's hard to compute. So what we can do is find this area over here to the left of P of Z equals 1.7 and subtract that from one. So after that, we get a P value of 0.0 for four. You're a 0.44 So for a part, ay, our answer is P equals 0.44 And now we're asked to test this Against an Alfa of 0.5 Our P value is 0.440 point 044 Sorry, because 0.44 is less than our alphabet 0.5 We can reject no hypothesis. So does that mean there's sufficient evidence to support the claim that sample one has a larger proportion in the sample?

In a problem. Eight. We're going to be comparing the number of games one by each league. We have two legs here the American League and the National League for Baseball games for the years 1970 to 1993 at the 0.5 level of significance, We're going to see whether there is sufficient evidence to conclude that with a difference in the number of kings. So the first step is to state the non Harper. This is then get the critical values wrong. The data then one called the values that to make a decision, whether to reject or real, to reject the night policies. So in our case, the null hypothesis change not is that there is no difference in the number wins and the alternative put. This is a case that there is a difference in the number of wins and both cases comparing the theme to leaks. But next we just did the critical Bali Andi, In our case, the critical value at 0.5 level of significance is plus or minus 1.96 Next, we're going to give the ranking full each a win, and he had the rankings for the American League and for the National League. So after getting the ranking, we need to find out the sample size of a small A sample. And in this case, the smaller sample is for the national legs. Onda. We have 11 wins and 11 samples and then for the American League, we have 12. So we're going to get Are we going to sound the rungs for the National League? So it's going to be 2.5 sound all the way out to 22.5 and the sum is 125. Next, we get Newmar from substituting the values off anyone and entered into the formula. And that yields 132. You are is obtained by substituting evolution to this formula, and we get 16.24 the value of Zell calculated values that is going to be obtained from the substitution of the values that we have just obtained. That is 120 five minus 132 divided by 16.24 And when you want that, don't get negative 0.43 08 now we can compare that one, but you off that to the critical value offset and the negative. Negative. 0.43 08 He is great. German negative. 1.96 If you can sketch the distribution, you find that the critical value is negative. One 1.96 And this is the critical region. While I watch critical, our calculated value is zero negative 0.4. So negative 0.4308 does not fall in the critical region and hens we make the decision to feel to reject. Then my life with this is And in this case since we've failed to reject the knowledge by this is we can't conclude that these not enough evidence to reject the claim that there is no difference in the number winds in the tunings.


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