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Hypothesis Tests About Proportion/MeanDefine the population parameter or distribution about which hypotheses are t0 be tested:The Null Hypothesis (Ho): 0 =The Alter...

Question

Hypothesis Tests About Proportion/MeanDefine the population parameter or distribution about which hypotheses are t0 be tested:The Null Hypothesis (Ho): 0 =The Alternative Hypothesis (Ha): M < M > p # P > P =p =Significance level:0.05The Test Statistic b: z = Pal_PaX - Lo a. t = s/Vn Observed Value of the Test Statisticp-value and decisionState the conclusion in the context of the problem

Hypothesis Tests About Proportion/Mean Define the population parameter or distribution about which hypotheses are t0 be tested: The Null Hypothesis (Ho): 0 = The Alternative Hypothesis (Ha): M < M > p # P > P = p = Significance level: 0.05 The Test Statistic b: z = Pal_Pa X - Lo a. t = s/Vn Observed Value of the Test Statistic p-value and decision State the conclusion in the context of the problem



Answers

Consider the hypothesis test
$H_{0} : p_{1}-p_{2} \leq 0$
$H_{\mathrm{a}^{*}} p_{1}-p_{2}>0$
The following results are for independent samples taken from the two populations.
$\begin{array}{ll}{\text { Sample } 1} & {\text { Sample } 2} \\ {n_{1}-200} & {n_{2}-300} \\ {p_{1}-.22} & {p_{1}=.16}\end{array}$
a. What is the $p$ -value?
b. With $\alpha=.05,$ what is your hypothesis testing conclusion?

To this lesson in this lesson we'll find the critical values for The instances to have three of them. We have the know hypothesis as the I mean just go to five as against the alternative hypothesis us the mean less than five. So less than five minutes. We have a lower to test to use the one tilt this approach we have enough of your point yo one and a simple size of swing. So we are looking for Quit cow value. So 0.01 With degrees of freedom of 19. Okay so 19 is here And now this is one till So there are 2.01 0.01 is equally here. So does the column. That's the rule. This becomes the figures of 2.539. You have 2.539. Okay let's go to the B. Part where we have enough of. Gerald point 05. And a sample size of 12. So we are little nut 2 0.05 And 11 as the degree of freedom. Yeah So 11 is here. 0.05. 0.05 is also here. So it comes this way and now we have This value of just 1.796 1.7 96. The c pad we are looking at I feel which is 0.1 And the sample size which is 15. So that s. t. and 14. The degrees of freedom. Mhm. Mhm. So their point one that is yet and 14 just so it comes this way and that So we have one 345, 1.345. Okay, since the time this is the end of the lesson.

Right When you take a sample or samples, the whole point is to use those samples to estimate the population proportions. So when you do, when you're using P. Hat one, that's your sample proportion from your first population. That's to estimate The population proportion of Group one. Likewise, P. Had two is going to estimate the population proportion of your second group or your second population and your PFP that's your pooled sample proportion. That estimates the combined population proportion Of your two groups.

When you take a sample or samples, those samples are used to estimate your population proportions. So when you have when you get an uh sample proportion from group one, that's going to estimate the population proportion. Mhm. From your first population or from your first group. Mhm. Yeah. And then your, he had to estimates the population proportion of group two or your second population. And then PLP is your your pooled sample proportion or combining your two P. Hats together. Which is going to estimate the combined population proportion of both groups? You should go. Mhm.

For this problem. We are given the following claim machines suspense amounts with standard deviation greater than 0.15 oz and we have the following information off to the side. So the first for this hypothesis tests the state or no hypothesis. So each not. We're talking about standard deviation. So sigma And it's gonna be our 1.5 oz are alternative since the claim is stating greater than it'll be sigma, Somebody greater than 0.15. So once we have our go on alternative hypothesis, let's find our testes cystic. Since we're talking about sanity, aviation, we're gonna be using the chi square distribution and the chi square testes cystic can be found by and before end minus one times as squared divided by sigma squared. Yeah, So this is going to be 27 -1. Yeah. Time 0.17 Squared Divide by 0.15 Squared. And the chi square tests should be approximately equal to 33 points 396. So I was the testes cystic, you can do the p value. There's two ways to do the people, you can either do since the sample size 27 we have a degrees of freedom. 26 But in the end -1. Since this degrees of freedom is relatively small, we can find it within the table or we can find the inequality arranged within the table. And since this is a right to test and the high score table, I'm using finds area right to the tip. Other critical values what we wanna do. We don't want to go down the degrees of freedom. Do we get to 26? We want to find the range where 33.396 is between the two numbers. So I found that is between 17 point 292. And it's gonna be less than 35 point 563. So now if we go all the way up we'll find that probably are the area to the right of the curve. So the 17.292 equivalent to 0.935.563 equivalent to 0.1. And now we have our P value. So RP values between 0.9 or 0.1 and 0.9. A more accurate where you can get your P value though is using software is like our you use the Arco p chi square which does it take the input of your chi square testes, cystic degrees of freedom and then lower dot L. Equals true or false. So the lord dot tails telling us whether it's a left tail test or right tail test. Since the right tail test, you want that to be false. If you plug in this information we should get to this p value is about 0.1 509. Which again agrees within the table method table inequality. So if you have the p value can make a conclusion by comparing your p value. What's your alpha. But you can also find your critical values. To find your critical value of critical values. Your again going to your chi square table is gonna be Since we're doing air to the right we just want our alpha. And so we are degrees of freedom. Remember that our alpha is 0.05. Our degrees of Freedom 26. So if we go to our high score table we can find were those two intersect and are critical values. 38 .885. All right. So, you can make conclusion with the critical values and your testes cystic. If you have a chi square distribution looking something like that, you're critical value. The states here Just 38.885. Okay, so you're critical value. We will reject any or don't have process. If it falls within this blue region, this is also called our rejection region. And if it falls anywhere within the non stated region, we will accept it. Which is also called our acceptance region. We see our test statistic as 33.396. So if you try to play set somewhere here that maybe belong around over there. Yeah. So that's archive square test. Yeah. And it falls within our acceptance region. So in this case for comparing test statistics and critical values we fail to reject If you want to compare with the P value and alpha. So a p value is greater than alpha. We failed to reject. H. Not. Yeah. and if P value is less than alpha, we reject H. Not again, we can look really quickly. We have AP value of 0.1509. And after a value of 0.05, our p value is greater than our alpha, so we failed to reject the null hypothesis. So in both cases we came to the same conclusion, which was we fail to reject H not.


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