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In 2019, statistics students in Quebec Universities scored uo=75.91 on average on a test of basic math ability. In 2020, the meanscore on the same test for a random...

Question

In 2019, statistics students in Quebec Universities scored uo=75.91 on average on a test of basic math ability. In 2020, the meanscore on the same test for a random sample of 20 Wuebec statiticsstudents was m=80, with a standard deviation of s=15.1415. Computethe lower limit of the approximate 95% confidence interval aroundthe estimated effect size computed with m-uo in the numerator (4decimals)

In 2019, statistics students in Quebec Universities scored uo= 75.91 on average on a test of basic math ability. In 2020, the mean score on the same test for a random sample of 20 Wuebec statitics students was m=80, with a standard deviation of s=15.1415. Compute the lower limit of the approximate 95% confidence interval around the estimated effect size computed with m-uo in the numerator (4 decimals)



Answers

Find the $90 \%$ confidence interval for the difference between two means based on this information about two samples. Assume independent samples from normal populations.$$\begin{array}{lccc}\text { Sample } & \text { Number } & \text { Mean } & \text { Std. Dev. } \\\hline 1 & 20 & 35 & 22 \\2 & 15 & 30 & 16 \\\hline\end{array}$$

For this problem. We need to find how larger sample is needed in one. A. Well, uh, this tender deviation Cocula notices. So that would be 2% 1 the table, too. Three. So let's try one for you. Even the head signal. Okay, find those suspects. Those signal because school defend yesterday that can be fired both sides by signal. So squirt up and close Ana every square about sand. Do this sample size the hands right for you. 40. So we need counter there for you. It's That's the

For this question we've been given and it's 24. The standard deviation given toe a sample standard deviation is 2.3. Since then, it's 24 degrees of freedom is 23. And since we're finding a 90% confidence of Derval Alfa is 0.1 these now we find that too critical values from the G table guy score alphabet to becomes Thais for 0.5 degrees of freedom 23 changing the G demon. This value is 35.17 Concise No guy square one minus Also by two Is guy swears their point Nightlife for degrees of freedom. Great three Checking the G table. This value was hurting points your 905 Now we have the critical values we need. Now we can move on to finding the confidence interval for the population variance. Do you know that we used this formula substituting the values we have? We get this right here. No, if you simplify this if you simplified the expression we have here to get the confidence in terms of Asians, you'll see that the confidence interval you get is 3.4592 as the lower boundary and nine point do 945 has the upper bonjour. Now this is the confidence interval for the population variance Now, to find the confidence and terrible for population standard deviation, we take the square group off the confidence interval we have for population variance. Let's do that. So we get square root off 3.4592 is the lower limit, which is less than people do the population standard deviation which would be less than equal to the upper limit which is square root off 9.294 fives sold this You see that the boundaries for conference and verbal for population standard aviation are one point it slitting fix those 1.8599 as the lower limit and the opportunity guest is three point zero for eight seven. So here we found the confidence interval for the population millions and the population standard deviation

In this question, we're told that a university entrance exam has average scores of 900 and a standard deviation of 180. So you can start by writing that down population mean as 900 population. Standard deviation is 180. And we're also told that every year the university's samples freshman to determine if the mean score on the entrance exams has changed. So for part A were supposed to produce hypotheses to test this. So then no hypothesis be that mean hasn't changed or that the mean is equal to 900. And the alternative is that the mean is not equal to 900 and then in part B, whereas to estimate a 95% confidence interval. And we're given that a sample was taken of night of 200 with a sample mean of 935. So for 95% confidence interval, Alfa is equal to one minus 0.95 and therefore Alfa is equal to 0.5 And so the confidence interval is like this. It's the sample average plus or minus critical values, said Sub Elf. Over to times the population standard deviation because we're given that divided by the square root of the sample size, and Zed's about for over two. You can look that up in the said table or you may have memorized it for Alfa equals 0.5 But that comes out to 1.96 and then we just substitute our values into the confidence. Interval equations with 935 plus or minus 1.96 has 180 squared of 200. And that gives us a confidence interval ranging from 910.5 2 959 0.95 So we can see where 95% confident that the troop population mean of freshman student in the year that sample was taken lies within this interval. We'll see, says, use the confidence interval to conduct the hypothesis test so we can just simply see if the population mean of 900 exists within the confidence interval and we can see that it does not so. Therefore, we can reject the no hypothesis for Part D were asked to do the same test, but using the P value instead. So to find out RPI value, we must determine our test statistic so there is equal to you. Sample average minus the population mean divided by the population. Standard deviation over the square root of the sample size, and that comes out to 2.75 And using the head table then is that score of 2.75 or a test statistic of 2.75 implies that RPI value is equal to 0.6 And since the P value of 0.6 is less in the Alfa Value or Alfa level of 0.5 we once again reject the no hypothesis.

And question number 21. The central limit theory tells us that the sample mean is normally distributed with me so very equal to 498. And this Thunder division, equal to signal over screwed offend equals 116 over square root of 20 all approximately equal to 25.938. And that the answer for this question. Thank you.


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