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09study was conducted to determine if the salaries of librarians from two neighboring states were equal: A sample of 200 librarians total were sarpled from the two ...

Question

09study was conducted to determine if the salaries of librarians from two neighboring states were equal: A sample of 200 librarians total were sarpled from the two states, half of which from one state and the other half from the other state The mean salary for librarians from the first state was S29,100 with a standard deviation of S2300. The mean salary for librarians from the second state was S30,500 with standard deviation of S2100. Test the hypothesis that the salaries for librarians between

09 study was conducted to determine if the salaries of librarians from two neighboring states were equal: A sample of 200 librarians total were sarpled from the two states, half of which from one state and the other half from the other state The mean salary for librarians from the first state was S29,100 with a standard deviation of S2300. The mean salary for librarians from the second state was S30,500 with standard deviation of S2100. Test the hypothesis that the salaries for librarians between the two states differ. Use a 0 = 0.05 level of isignificance Write out the hypotheses: Detennine the critical values: Compute the value of the test statistic: Determine the decision: Write the concluding statement:



Answers

Perform each of the following steps. a. State the hypotheses and identify the claim. b. Find the critical value(s). c. Find the test value. d. Make the decision. e. Summarize the results. Use the traditional method of hypothesis testing unless otherwise specified. Assume that the population is approximately normally distributed. The average cost for teeth straightening with metal braces is approximately $\$ 5400$. A nationwide franchise thinks that its cost is below that figure. A random sample of 28 patients across the country had an average cost of $\$ 5250$ with a standard deviation of $\$ 629 .$ At $\alpha=0.025,$ can it be concluded that the mean is less than $\$ 5400 ?$

Question 27. We're told that the weekly average unemployment benefit in the US is $238 and we're also told that a researcher in Virginia state suspects that the state average there is a less than $238. So for part A were asked to develop a hypothesis test for this situation. So we can describe the Virginia state researchers view as the alternative hypothesis that Mu is less than 238 and therefore the no hypothesis is mu is greater than or equal to 238. So that's your hypothesis test for Part B. We're told that we have a sample from Virginia State of size 100 and the average of the sample is $231 and the sample standard deviation is $80 and we're asked with the T values, so we're not given the population standard deviation. So we can say that sample averages are distributed according to the T distribution, so tea is there test statistic, and that is equal to the sample average minus the population average over the sample standard deviation divided by the square root of sample size and that comes out to minus 0.88 and then were asked to calculate the P value. So just to show you what's happening here, so this is the T distribution. It's symmetrical and this is a lower tail test because the alternative hypothesis suggests that the mean is lower than the no hypothesis. So basically we have a test statistic that comes out to minus 0.88 and we're basically asking the question. Does it fall in some critical region in the lower tail such that it's low enough or extreme enough to reject the no hypothesis? So the critical region would it be defined by Alfa, which we haven't been given yet? And so the P value is the area to the left of our test statistics of the red area, so it amounts to comparing the P value to the Alfa value to see which is smaller. The P value of smaller than our test statistic is more extreme and lower in, which would lead us to rejecting the no hypothesis. But of course, the tea table gives upper tail values. So we're looking at areas in the upper tail but our test statistic if we just take the opposite of it. So plus 0.88 we would find their the area that we're looking for. So this is also equal to the P here yet, so these two tails are identical, so we can go to the table and use plus 0.88 to find RPI value. But one thing we need before we go to the table is the degrees of freedom, and that's simply and minus one, and that comes out to 99. So then, when we go to the tea table, we want to roll. That is 99 degrees of freedom, and our test statistic of 0.88 falls between these two values, which means that our P values between 0.1 and 0.2 so that's the T value. So Part C says at an Alfa equal to 0.5 what do you conclude? And we can simply say that the P value is greater than Alfa for this situation, and therefore we failed to reject the no hypothesis then for party, whereas to repeat the hypothesis test using the critical value method so we have to find her critical value. So Alfa is 0.5 and the critical value then has gotten from the table. So again it's 99 degrees of freedom. Upper tail area of 0.5 corresponds to Alfa and therefore critical values 1.660 But remember that this is a lower tail test, so it's actually minus 60.66 three. Let me just check that daily again minus 1.660 So on the left tail, critical value is equal to minus 1.66 and then to use the critical value method, we simply compare our test statistic to the critical value. And our test statistic is equal to from part B minus 0.88 and that is bigger than minus 1.66 which is the critical value. And therefore, in this case, we also feel to reject the null hypothesis

Okay, let's look at question 14 together here. So, um, our no hypothesis ISS The median is equal to 83,000 and two under 82. And the alternative hypothesis is that the media and is smaller than this amount of money. Um, because here, the sign is smaller than so, which means we're looking at the one talk situation. All right, um, by looking at the table list at the low, we know that the sample size is 20. And, uh, and off Linus is 16 and off, plus is four. So our X star that is going to be, um, is going to be four, cause for a smaller than 16. All right, um, So what we're going to do right now is that we're going to look at table 12 and find the corresponding line for Enos. He go to 20 so Oh, just try here. Can the above, Because their own point. There are one narrow mind. No, by their white one 0.2 by and we have three high by six and know that here are Crippen. Our X star is four. So four is in between three and five. So which means our P value should be tweeting between their open, their one divided by two here and smaller than their opine. There are five D bye bye to the result women to divide by two year. Because instead of looking at to tell situation, we're actually looking at one type situation. So we need Teoh, um, divided by two and which is going to be, um, people greater than their A 0.5 and smaller than the aeroplane. There are 25 And from the question we know that the ec Tropea Vallo calculated by the computer is Geico to their A pine. There are there, 05 night and is in between the pounds. We just find out. So, um

22 it's note that the mean is one of them are equal toe 18,500 and it's one all the alternative hypothesis city event that mean is bigger than 18,500 for question be that the T value is equal to X bar minus meal. Note. Over S over square Root of act, which is 20086.5 minus 18, 500 over 1417.739 a nine over square root off 12, which is equal to 3.876 for questions. See the critical value can be deter mined you think they will find with degree of freedom equal toe end minus one. She's 12 minus one, so that's equal to 11. And the column equal toe offer is equal toe. Open it 11 tail so T is equal to 1.363 So the rejection reason it then below this T value. So if the T value is introduction reasons, there's no hypothesis is rejected at 3.876 is bigger than 1.363 So we reject then on high processes. So which, and we say that there is sufficient evidence to support the clean

18. So it's no if that new one is equal to mean to, and each one is that anyone is not equal commute. So the degree of freedom, the minimum off in one way, this one, which is 17 and 18 to minus one, which is 19th. So the video is 17, using all for April toe 0, 00.11 teen. So for table five, the critical, very equal toe postive or negative 1.74 So the rectory, then they contain all value smaller than negative 1.74 and all various larger than 1.74 So that is the statistic, if that makes one minus x two bar. So it's 56900 minus five 7800 minus me one minus me, too over square root off this one squared over in one, which is one toe. Once you lose your square over and once with the 18 lost, it's too square. So it's 8000 square over into approximately negative 4.267 So it's a very off statistics. Even in the final hypotheses rejected so and this value realize between the booze values. So we failed to reject the null hypothesis


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