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2. (28pts) Identify the_critical values which define the rejection regions for these alternate hypotheses, which may or may not have two tails (2 decimal places):Us...

Question

2. (28pts) Identify the_critical values which define the rejection regions for these alternate hypotheses, which may or may not have two tails (2 decimal places):Use t~-distribution: (Zpts) Ha: p # 150,n=80, 0-0.10 b. (Zpts) Ha: / < 40, n=10,0-0.01left tail: left tail:right tail: right tail:Use Z-distribution: (Zpts) Ha: 4 >-0.10,n-20,&-0.20 d. (Zpts) Ha: H # 6.5, n-40,0-0.04left tail: left tail:right tail: right tail:

2. (28pts) Identify the_critical values which define the rejection regions for these alternate hypotheses, which may or may not have two tails (2 decimal places): Use t~-distribution: (Zpts) Ha: p # 150,n=80, 0-0.10 b. (Zpts) Ha: / < 40, n=10,0-0.01 left tail: left tail: right tail: right tail: Use Z-distribution: (Zpts) Ha: 4 >-0.10,n-20,&-0.20 d. (Zpts) Ha: H # 6.5, n-40,0-0.04 left tail: left tail: right tail: right tail:



Answers

Find the rejection region (for the standardized test statistic) for each hypothesis test. Identify the test as left-tailed, right-tailed, or two-tailed. a. $\quad H 0: \mu=141$ VS. Ha: $\mu<141$ [email protected] \alpha=0.20$. b. $\quad H 0: \mu=-54$ vs. Ha: $\mu<-54 @ \alpha=0.05$. C. $\quad H 0: \mu=98.6$ VS. $H a: \mu \neq 98.6 @ \alpha=0.05 .$ d. $\quad H 0: \mu=3.8$ VS. Ha: $\mu>3.8 @ \alpha=0.001$.

All right. So in this question we are looking at determining the rejection region. Um for the test statistic also what tail it is. So for the first one we're testing against μ not equal to negative 62 at a really small significance level of .005. Which in this case um since mu is not equal to that is going to be a two tailed test Which means that that 5% probability is contained within two tails. That means we need to divide it by two to find the position. Half of that rejection probability is here, half of it is here. Um So when we divide that into We get .0025 and scrolling down to the infinite degrees of freedom on our Figure 12 3 and the Appendix Less than -287 or Greater than or equal to positive 287. Okay. For part b You greater than 73. So one tell test At a small significance level of .001. It's um to the right tail because it's greater than that means we're gonna be on this side of our curve and we don't have to divide our significance level into because all of that rejection probability is contained in one tail. So .001 for the tail probability scroll all the way down Greater than or equal to 3.9. Part C Mu less than 1124. I'm sure you've noticed like this number really doesn't matter. It's really the sign that is the only thing we're interested in here, it will matter Once we start actually computing the test statistic, then that's when things will start to matter. Um Which is coming in number five for this section. So again we're looking at a one tailed test that is left tailed here and so it's gonna be less than or equal to the negative Z score. We're going to look for that 0.1 tell probability which is negative 3.9, same as what we had in part B. And finishing things up in part D. We are doing a two tailed test because we're looking at not equal to 12 for our main With the significance of .001. So it is two tailed because of the not equal to. So we have our negative test, we have to split that tail probability in 2.001, split into two as .0005. And the Z score at the bottom there is 3.291. So that wraps up how to determine the rejection region. And then as we move on we will start looking at what things look like when we are actually finding the Z score

Alright so again we are determining the rejection region based off of what the um Significance level is and whether or not it's a two tailed test or a one tailed test. So the first one we're testing if it's less than 17 With a significance level of .01. So that's a one tailed test. We're going to go to our appendix and look at the 1% probability, the tail probability of .01. Since it is a negative or less than 17 we're gonna take the negative Z. Score of 2.3 to 6. Okay Moving on to part B. We are testing if it is not equal to 880. So we have our significance level of point a one. Since this is a two tailed tests we have to divide it by two to figure out where it's going to be here because again the 1% is gonna be contained within both tails not just within one. So that's .005. Um which is 2.576 on the Appendix in the back of the book. So anything less than or equal to negative 2.576 or greater than or equal to the positive Z. Score that matches there part C. Um or Looking for our mean to be greater than -12 with a significance level of .05. So greater than is just a one tailed test. So we're looking for a Z score greater than or equal to um the 5% tail probability gives us a Z score of 1.645. An easy score greater than that would be grounds to reject they know. And then d we are testing against another two tailed tests 21 point one And our significance level is .05. So we're gonna divide that by two which is .025 is the tail probability Which is 1.96. So anything less than negative 1.6 or greater than positive 1.96. So the key here is using that figure 12.3 to be able to determine that and noticing if it's a one tailed or two tailed tests to know if I'm gonna split the tail probability into um to find the location on the density curve. Or if I'm just going to take it as it is. If it's a one tailed test like A. N. C. We're here.

The null hypothesis for this problem. Bilby, it's not sick must very called to 25 and sensitise. Given that this dough pend best, the alternative hypothesis at June will be SIG must quit not equal to 25. So these are our no reliable basis and alternating foetuses. Now we have to find that critical values. First we find the degrees of freedom degrees of freedom equal to n minus one and is given Toby 14. So degrees of freedom equal to 14 minus one, which is equal to talking. Since the Testes Hotel, the area must be spirited. I'm fine with 0.10 so far divided by two equals two 0.10 divided by two, which is equal to C 0.5 So 0.0 over will be the right sex Shadia and 0.0 I will be the left sex shaded area for life's like Katie can venue. We will see in the table for guys car distribution under degrees of freedom, Torti. And I'll fight while to 0.5 so the like sex critical value is 22.362 as seen in the table forecast square distribution far left such critical value. See the value under a fine well one. Minus this. 0.5 So one minus 0.2 Your five is equal to 0.95 Soc undersea open 954 degrees of freedom 13. The value is 5.89 Do as seen in the table for the chi Square distribution. These are the quite critical values pipe on a plan toe and 22.362 30 The region right side to the life sex critical value that this this shaded region is are critical region. Also the region toe the left off the left. Such critical value is our critical region that this these two shaded regions are our critical regions. So the unshaded region is our noncritical region. Now, for this I will not happen. Texas will be seen that there's no Apatow cystitis. It's not Sigma squared equal to 25. The alternative hypothesis actually will be as it is, Like that test a 20 basic must square greater than 2 25 So these are not hypothesis and alternate new hypothesis is now the grease off freedom equal toe end minus one. So degrees of freedom equal to 27 minus one That this 26 for critical value Since it this right third test, See the value under I'll fight while do 0.5 which is the given value off and degrees of freedom. Equality 26. The critical value is 38.885 as seen in the table for the chi square distribution the region. So the right side off the right side critical value that is this is that this this shaded region is are critical region. So the uncharted region is the non political future. In this problem, toe null hypothesis will be saved that this it's not will be Sigma squared equal to 25 as it this next big test our identity. Our POTUS is a joint will be Sigma squared less than 2 25 So these are not hypothesis and alternative. High potency is now the grease off Freedom IQ were to end minus one and has given Toby nine. So degrees of freedom equals nine minus right, which is equal to it. I think. What was 0.0 and take this This region is 0.1 We can now find the critical value. Sensing this lefton best for critical value V V C under one minus this. 0.1 So one minus 0.1 which is equal to 0.99 That this we will see and does your 0.994 degrees off freedom equal to eight. It is but like 646 as it is White Park 646 as seeing the table for Chi square distribution the region do the left off the left sex critical value is our critical region. That Mr Shaded Region is a critical region. So the incident region is our noncritical region now and still our nine hypothesis edge not will be sick must quit equal to do Banky since it this right thing best the alternative high POTUS is joining Will be sick must quit greater than 2 25 degrees of freedom The yes equal tow n minus one and SG one Toby 70 So degrees of freedom equal to 70 minus one which is equal to 16 on advice given to be 0.5 that this well this shaded region is 0.5 Since it doesn't like think best see you under a cycle to 0.54 degrees of freedom Equality 16 in the table. For chi square distribution, the required value is 26.296 which is our critical value. The region to the right side, off the right sex critical venue that this hour shaded region is our critical region. So the luncheonette region is so what no critical region.


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