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17. The teacher leamed ofa new prep class, ACT Now They want to know students scoresfrom this prep class differ from the scores from the students who took the ACT U...

Question

17. The teacher leamed ofa new prep class, ACT Now They want to know students scoresfrom this prep class differ from the scores from the students who took the ACT Up prep class-What test should be conducted?State the null and research hypotheses in symbolic notationCalculate and report the test statistic In APA format,Make A.dectsion (rejectfail to reject the null) and communicate the results (what do the results mean?)Calculate confidetice mntervalCalculate and interpret the effect size

17. The teacher leamed ofa new prep class, ACT Now They want to know students scores from this prep class differ from the scores from the students who took the ACT Up prep class- What test should be conducted? State the null and research hypotheses in symbolic notation Calculate and report the test statistic In APA format, Make A.dectsion (rejectfail to reject the null) and communicate the results (what do the results mean?) Calculate confidetice mnterval Calculate and interpret the effect size



Answers

For Exercises 5 through $18,$ perform these steps. a. State the hypotheses and identify the claim. b. Find the critical value. c. Compute the test value. d. Make the decision. e. Summarize the results. Use the traditional method of hypothesis testing unless otherwise specified. Assume all assumptions are met. Extending the School Year A researcher surveyed 100 randomly selected teachers in a large school district and found that 46 wanted to extend the school year, 42 did not, and 12 had no opinion. At the 0.05 level of significance, is the distribution different from the national distribution where $45 \%$ wished to extend the school year, $47 \%$ did not want the school year extended, and $8 \%$ had no opinion?

In Problem 22 told that the overall U. S. Public health tradition high school graduation rate in 73.4% and for two states, Pennsylvania and Idaho it is 83.5% and 80.5% respectively. And you can see the difference is only 3% tool that random samples of 1200 students from its it sure that 980 from Pennsylvania graduated while 940 from Idaho graduated. So we're going to be testing the clean. There's a difference in the proportion off graduating students between the states, and we will do so using the 0.5 level of significance. So the first step is to state the they're not hypothesis on the alternative hypothesis. So the number of this is his P one equals P two. The alternative before this is is p one is not equal to be to, and this is the claim that we agreed to test. Next, the level of significance is 0.5 and since this is a two tails test, the critical Nadhim is plus or minus 1.96 Now we can work out the value of P one hunt. That is the proportion off students who graduated in Pennsylvania. And it's given by the Fraction 980 divide by 1200 and that in decimal phone is zero point h 17 Next p too hot. The proportion of students who graduated in i down it's going to be 940 divided by 1200 and decimal form 0.783 Next, we cannot caught PBA by substituting the Bradleys, UM x one x two and one and into. So that's going to be 918 plus 914 divided by 1200 plus 1200. And when we simplify, PBA equals 0.8, Cuba will be one minus 0.8, which will be 0.2. So with this mindless, we can walk out the test of you by substituting the bodies of obtained into the formula for that. So that's going to be 0.817 minus 0.783 minus zero. She's there assumed different in proportions for the population divided by the square it off P bonds. Your 0.8 times Cuba 0.2 times The fraction one off N one is one of the wonders of 200 plus 1/1 dollars and 200 which is to. And when we simplify, we'll get that. The calculated value said is 2.0 for one. We now need to compare the completed bundles there and the critical value upset. So we find that the calculated volumes at 2.0 for one years great happen the critical value, which is 1.96 This means that the progressive Vandy offset is within the critical region, and when that happens, we make the decision to reject the now like Oh, this is and rejecting the knowledge that this is leads us to make the conclusion that there is enough evidence to support the claim that the proportions are different. So we have enough evidence from the sample to support the clean that the proportions off graduating students between the states are different.

So in this problem, we going to test if the average number of application is greater than 7.8 Celeste Data hypothesis. Asian artists mu it go to 7.8. Chuan is me you greater than 7.8? So this is our claim wishes, right? Ill test in this problem. So for a significant level, Alfa is your poisoner one that gave us a positive critical value at C ego to 2.33 9 Next, we're going to calculate the test value, which is the ego to a 27 minus seven point 8/2 7.0.6 Always were 35. So this is too boyo. Fight. So let's look at this crap. So we have the critical region. Iss Alfa is poi a one that give us a critical value of 2.33 our test value is 2.0 fi, which is less stand the critical value. So at this region, we not reject h not so we can call that there's not enough evidence to support our claim, which is the original member application is going toe than 7.5 point eight at the significant level. ALF I 0.1

In problem 15. We wanted to test the effect of a new course for the preparation for the mathematics section of the state. Exam 20 which means N equals 2020 students is recruited for the course, and their average score after taking the exam is 562 which means M equals 562. The general population had I mean of 500 and Sigma standard deviation equals 100 party. We want to see if that this has a significant effect on the meaning score for the students. Let's first declare our hypothesis. They're not hypothesis, Cesar, you equals 500. It doesn't change by the effect of the new course. And of course, the alternative hypothesis will be, um you will be greater than 500 500. This is the alternative hypothesis, and we will use Alpha equals 4.1 four. I want the old test. This is the first step is to clear the hypothesis. The second step is to get that's that's cool. But before we get the score, we get Sigma M. Sigma M equals the standard division divided boy square root of the sample size, the central division of the population is given by 100 divided by the square root of the sample size sample size given by 20 which equals 22 points. 36. This is the standard error, or sigma. Now we can calculate the score they score equals M minus mu divided by Sigma M Mm. It is estimated by 562 from the symbol minus mu is 500 divided by Sigma M 22.36 This gives 2.77 Now we want to get the area to the right of the city score. We have the normal standard, the standard normal distribution tables which gives the area to the left over the school, which means to get the B value. It equals the area which is one minus the area to the left of the score. We can get the area to the left of the score from the tables. As we have said, let's enter the tables with the equals. 2.77 2.77 is here We have here 2.7 and 987 Then we have here the area to the left of the score. It's all 0.99 72 4.9972 Which means the B value equals Oh boy, or to it, which, of course, it's less than the value of Alpha were buoyant. Over this means we reject. It's not, which means there is a sufficient evidence to consider the course to consider the course significant. The course is significant or has a significant effect on the mini school for the set. This for Barnaby. We want to calculate Cohen's D to estimate the size of the effect. Coins deep just equals M minus mu divided by sigma. It equals 560 to minus 500 divided by sigma, which is 100. The population that's under the vision of the population, then it equals oh boy in six to, which means the effect is medium. There's a medium fixed because the media effect has a currency about 4.5. Finally, for board seats, we can use the results of A and B together to give a significance and and if it conclusion, the test or the experiment makes the experiment has concluded that the new course has a significant and medium if it on the results of the S 80. Well, they said it's significant and with a medium effect, and this is a final answer of our problem.

All right, so in this information, we're given some a C T. Summary data from 2000 were given both the mean on and the standard deviation from a sample of English and math tests. So the English, I mean, was 20.6 with a standard deviation of six. The math mean was 21 with a standard deviation of 5.1. And in order compare these. We're gonna need to know the formula for the Z score X minus X bar over s where X is the data value expires. The mean and s s a standard deviation. So part A what if somebody scored a 30 on which of these exams the English or math? Would they have performed better relative to that subject? So for English will substitute the values into our Z score formula, and we find that this has a Z score of 1.56 So this score of 30 it's 1.56 standard deviations better than average for the English test for math. We repeat the procedure using that's statistics from the math test, and we find that this person are that this score is 1.76 standard deviations above the mean. So who had the better performance? Um, relative to the English and math scores. A score of 30 indicates a better performance in the math on the math test, right? What if we had a score of 23 so we'll roll. Repeat the same procedure 23 minus 20.6 over six gives us a Z score of 0.4. And on the math test, 23 minus 21 over 5.1 gives us a Z score of 0.39 So I would say that for these two, um, fields these two subject areas, though, z scores, but as close as you can get, So the performance is about the same. And then we're gonna look at a lower test score. So his test score of 12. So on English, we have 12 minus 20.6 over six. So since it's going to be below average, it's gonna have a negative Z score. So negative 1.43 And on math, we're also gonna get a negative Z score. So negative 1.7 So in which subject dairy was the performance better and in this case, it was on the English test because it is fewer standard deviations below average than the map. So the next part of the question says, Then why did the better test switch from English to math Between questions A and C. So why was 30? Why was that so much better than average And why I was 12? Um, I was 30. So much better in math than average. And when I was 12 so much better in the below average range. So why did this switch from math English? And it really has to do with, um, the relative nous to the standard deviation. So 30 is more standard deviations above their means in math, and 12 is less standard deviations below the mean in English. So since what we're measuring a standard deviations, that's what we're looking for. So this is closer to zero. That means it's better because a score above zero is a better score. So this person is far, but are further below average than this person. So the map is further below average than the English. All right, And then part e, we're gonna look at one specific student who scored a 26 so our data value is 26. So did that student do better on the English tests, or did they do better on the math test? So we'll calculate the Z scores in the same way. So for English, the student would have a Z score of 0.9, and for mouth student would have a Z score of 0.98 So even though it's pretty close, it looks like the student has a better performance on the math test because her score is more Sander deviations above the respective mean meaning. More standard deviations above the math mean that the score of 26 would be above the Englishman.


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