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The annual number of hits was compared for three baseball players 5 seasons were randomly selected (n = 5) and the following statistics were determined:Player L X, ...

Question

The annual number of hits was compared for three baseball players 5 seasons were randomly selected (n = 5) and the following statistics were determined:Player L X, =122.5 S1 = 25.1Player 2 72 = 161.2 S2 = 14.8 sz =Playcr 3 X3 = 137.9 S3 = 22.7Use ANOVA to determine if at least one of the population means of annual hits for the players is different from the others. Use & 5% significance level.

The annual number of hits was compared for three baseball players 5 seasons were randomly selected (n = 5) and the following statistics were determined: Player L X, =122.5 S1 = 25.1 Player 2 72 = 161.2 S2 = 14.8 sz = Playcr 3 X3 = 137.9 S3 = 22.7 Use ANOVA to determine if at least one of the population means of annual hits for the players is different from the others. Use & 5% significance level.



Answers


The following data represent the salaries of 14 randomly selected baseball players in the 2010 season (data are in thousands of dollars, so 3250 means $\$ 3,250,000$ ).
(TABLE CAN'T COPY) Test the hypothesis that the median salary is more than $\$ 1000$ thousand (\$1 million) at the $\alpha=0.05$ level of significance.

We have the following sequence of numbers starting at 7.1 and ending at 0.8. We have to we want to conduct a runs test and randomness on this data set. However, first we need to complete step one which is to convert our data to form a above the media and be below the media. I use the green orange differentiate numbers above and below the median. So we obtain median 8.35 which means that our sequence can be written in terms of A and B. Here. Next we have all we need to implement the runs Tessa randomness so we proceed through such A three below and every state alpha hypotheses, this gives alpha equals 30.5 H not symbols are randomly placed. H A symbols are not randomly placed and be re compute and one and two. And are this gives anyone equal and equal six and two weeks every equal six. The runs as counted here gives R. nine. And see we calculate C one and C two from a runs test table for N one, N two. This game C one equals three, C two weeks 11. Thus we conclude Indie that since ours between C one plus one and C two minus one, we fail to reject H not which means that we lack evidence. The numbers are not random about the media.

All right, I'm going to use Microsoft Excel for this question. Um Let's see here. So we've got National League and American League and we've got a range 2.24-2.246. Um but it's different between National American American League. So um let me put in the numbers .24-2.246 uh .247 2.251 0.25 to 2.256 point 2572 point 261 .262 groups, 2.266. 2.271. So that's for National League. And I'm going to put en el above it. I'm gonna do the same thing for american League A. L .244, two 0255 256261 two 62 67 .268 273.274, Okay, so now I'm just going to take the average of each of the bins. That's going to give me an average here. And I'm just gonna pop you that and paste it down here. Okay, now there's a number next to each one, three players, 6 players, one player 11 11, 1, 362130. I haven't mentioned it. Okay, So now I want to take um the first column times the second column. Okay. And I'll just drag that the whole way down. Okay, I'm gonna some this So that means there's 33 players there and some this we just copied from above And 15 players here. Okay, so um what did I do with this? So I multiplied by three. Okay, this is going to give me the mean, because I'm going to some this. Okay, And then I'm going to divide by the 33, so the mean Is 25, 7 makes sense. And then down here, so I'm going to write mean underneath it. Give myself one more spot here, right, mean underneath it, and this All right, Count how many there are. Yeah, Okay, now the difference from the meat is going to equal this, minus The mean, and I'm going to press F 4 to keep that there. Oh, I must have clicked the word mean, I wanted it to be E seven, No E eight, Yeah, Okay, so distance from the mean. Um I'm gonna square that distance. It's gonna drag it the whole way down up. No, not gonna drag it to here. Now I'm gonna copy this, Paste it down here. This will be the distance from the mean here, which is at 17 doesn't seem right? Oh, squared I squared it. All right, so there's that's the square of the deviations, square of the deviations. Now, I'm going to take that and I'm going to multiply it by how many there are, mm. Just copy that may sit down here. Okay, so now I need oh yeah, this will be fine because this one ends up being zero anyway, because there are zero of them. Okay, so now I'm going to some this and I could just copy this right here. Not exactly some of the square of the deviations, and then I'll type it down here, some of the square of the deviations. So since this is a sample, I take the sum of the squares of the deviations, which is here, the sum of the squares of the deviations. Let me think here, yep, which is this? And then I have to divide by N -1, which is the count minus one, that's the variance. So this divide by the count, nope, abide by the count minus one. That's the variance. Okay, and then, so I'm gonna write variants here and then the standard deviation yeah, is just going to be the square root of the various. Okay, so we see that in this sample there's a greater standard deviation in the american league, but that's probably because there were fewer uh chosen. Okay, uh compare the results, that's what I just did. Um It's a little bit greater in the american league, thank you for watching

We want to conduct a hypothesis test dealing with some data collected from Major League Baseball World Siri's um instead of the chart going in a more horizontal direction, I'm gonna make it go in a more vertical direction. So sometimes world Siri's lasts only four games. Other times they last five games. Sometimes they last six games, and sometimes they last seven games. So the study was to take 105 Major League Baseball world Siri's and break them down by Was it over in 24 games, five games, six games or seven games, and the observed data for the contests were that 21 of those 105 games ended after game 4 23 ended after game five 23 ended after Game six and 38 ended after Game seven. And that is what we would refer Thio as are observed data. Now we have some expected proportions. We expect two out of every 16 to go on Lee four games. We expect four out of 16 to go five games. We expect five out of 16 to go six games and we expect five out of 16 games to go seven games and again. Our goal here is to run a hypothesis test. Well, in order to run a hypothesis test, we are going to have to construct, um, are null hypothesis. And our alternative hypothesis so are no hypothesis is going to be that the actual number of games fits the distribution by the expected proportions. Yes. And the alternative to that, or the alternative hypothesis would be the actual number of games does not fit the distribution mhm by the expected proportions. And in order to run this, we're going to be running a goodness of fit test. And to do a goodness of fit test, we're going to have to find a chi square test statistic. And the Chi Square test statistic is found using the formula. The sum of observed minus expected quantity squared, divided by expected. So we're going to have to go back up to our chart, and we're going to have to calculate the expected number of games, not just the proportion. So we want expected games. Well, if we are expecting two out of 16, then how Maney would that have been out of 105? So if I do my cross products, I'd get 16 X equals 210 so the expected number would have been 13.1 to 5. Then I'm gonna do the same thing for the 4/16. I would expect four out of every 16 games or world Siri's to go to five games. So how much is that out of the 105 that I included in this study? And if I cross multiply that I would get 16. X equals 420 and the result would be 26 point 25 And then I'll do the same thing for the five out of 16. So if we had five out of 16, that would be what out of 105. And again, if I cross multiply 16 X equals 525 and I get an expected value to be 32.8125 And that's the same for both lasting six games and or seven games because they had the same expected proportions. So we're ready to find that Chi Square statistic, and we're going to have to create a new column in our chart, and the new column is going to be called observed minus expected quantity squared, divided by expected. So in order to calculate those, the easiest way to do it is to put the information into our graphing calculator. So I'm gonna bring in our graphing calculator, and I'm going to clear anything that I might have in my list. So I know I've been using list one list to enlist three quite regularly, so I'm gonna make sure they're clear before we start. So it hits, stat, mhm and edit. And I thought I cleared out my lists, but let's try it one more time. There we go. So we're going to put the observed values into list one? Yeah. And we're going to put the expected values into list too. Yeah. Mm hmm. And we'll put the, um, stuff there. Let's try one more time. 32.8125 and then the final one. Okay, so we have all our data, so we want the calculator to give us information into list three. So we're going to sit on top of list three, and we're going to tell it to take all of the values, the observed values that we've placed enlist one, subtract all the expected values that we've placed Enlist to square that difference and then divide it by all the expected values enlist to, and we're going to get a variety of decimal answers, and I'm going to write them down in the chart, and I'm going to go out to three decimal places with each. But I'm going to keep the full decimal in the calculator. So we will have for the 21 games, will have four point 7 to 5 for the 23 games will have 402 for the 20 or for six games or the Siri's lasting six games, we'd get 2.934 and for Siri's lasting seven games, we would get 70.8 20 Now, remember what the Chi Square test statistic said to Dio. We need to add up these values, and the easiest way to add those values up is to bring our calculator back in, quit out of the list mode and ask second stat math. Let's sum up all those values that were enlist three. And in doing so, we get a Chi Square test statistic as a dust small of 8.8819 so our Chi Square test statistic is 8.8819 Now. The next part of the hypothesis test is to calculate a P value and to calculate RPI value. What we're really asking is, what's the probability that Chi Square would be greater than that test statistic? And I always like to draw a picture to kind of model that. So we know we are using a chi square distribution and chi Square distributions are generally skewed to the right, and they are based on the degrees of freedom and our degrees of freedom are found by doing K minus one. And K represents the number of categories that we have divided our data into. And if we go back to our chart, we see that we have broken our data into four different categories. How Maney of the Siri's lasted four games, how Maney lasted 56 and seven. So our K value is going to be four, resulting in our degrees of freedom being three. And that degrees of freedom also tells us what the average of the distribution is. So our average of this chi square distribution is going to be three and will always find our average slightly to the right of the peak. So down here on our horizontal axis are Chi square axis. We would find three in that location. Now to find that p value we're looking for. What's the likelihood of the probability that Chi Square is greater than this value? Well, that value would be further to the right from three, and we'd have eight 0.8819 and we're looking for the likelihood or the probability we're running greater than that. So this would be where the RPI value would be, and it's It's that area in that right tail. And the best way to find that area is to use the chi square cumulative density function. And when you use that function, it asks you for the lower boundary, the upper boundary and then the degrees of freedom. So in our instance, our lower boundary is the 8.8819 The upper boundary is found way out in that tail, and that tail, um, keeps going. So we're going to use a very large number. We're gonna say 10 to the 99th power and then our degrees of freedom we calculated to be three. So I'm going to bring in my graphing calculator again, and I'm going to show you where you can find that Chi Square distribution. So you're going to hit your second button and your Vares button and it's number eight in my menu. And again, it asks for that lower value. That lower value was 8.8819 We said our upper value is going to be 10 to the 99th Power, and our degrees of freedom was three. So our P value, where the probability that the chi square would be greater than 8.8819 ends up being 0.309 We're almost done with our hypothesis test. Now that we have a P value, it's time to make our decision. And we want to run this hypothesis test at a significance level of 0.5 And we describe our significance level utilizing the variable Alfa, and we will make the decision to reject the null hypothesis. If that's the statement, if your Alfa is greater than your P value, So in our instance are Alfa was 0.5 and it is indeed greater than RPI value because our P value was 0.309 So our decision will be that we're going to reject the null hypothesis. So let's go back up to that null hypothesis and we're rejecting it. So basically, we're throwing that statement away. And by throwing that statement away, then we're throwing our support toward the alternative hypothesis, so we can then make the conclusion that there is sufficient evidence right to conclude that the actual number of games played in a major league baseball world. Siri's does not fit the stated expected proportions. So we're saying that those

For Problem 72 were given a table representing two baseball players. Fatal and call were also given their batting averages, the team batting average and their team standard deviation, and we are asked to find the value. But it's three standard deviations above the mean on below the mean so in order files. To do this, I used the team putting our village and the teams down a deviation and, as you can see, yeah, 0.16 things was the team average was the team batting average for Fredo and a zero comma zero ones. Who was the team standard deviation for fatal. So to find the value that is three standard deviations above the mean you simply take the team batting average. And then we add 0.12 multiplied by three because we're trying to find the value. But it's three Spanish deviations. I mean, so this gives us an answer off zero point so zero to exactly the same thing for a girl, and you'll find an answer off zero points with the four. So the 0.189 was the team average was the team patsy are visual for call. And the 0.15 was the same standard deviation for Karl. Whoever wants a question be, they ask us to find the values but our three standard deviations below the mean So you do exactly the same thing. And the only thing that you change is the sign instead of a plastic we ever mine is now because you're looking for the values that I've been. No. The mean off 0.166 and 0.189 Calculating these, calculating these numbers over here you will find an answer off 0.13 and 0.144 Well, that was easy to understand, guys. Thank you.


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