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The table below Iists the number of games played in yearly best-of-seven baseball championship series, along with the expected proportions for the number of games p...

Question

The table below Iists the number of games played in yearly best-of-seven baseball championship series, along with the expected proportions for the number of games played with teams of equal abilities Use 0.05 significance evel to test the claim that the actual numbers of games fit the distribution indicated by the expected proportions: Games Played Actual contests Expected proportion 1 1 1Determine the null and alternative hypothesesHo: The observed frequencies agree with the expected proportion

The table below Iists the number of games played in yearly best-of-seven baseball championship series, along with the expected proportions for the number of games played with teams of equal abilities Use 0.05 significance evel to test the claim that the actual numbers of games fit the distribution indicated by the expected proportions: Games Played Actual contests Expected proportion 1 1 1 Determine the null and alternative hypotheses Ho: The observed frequencies agree with the expected proportions Hq: At least one of the observed frequencies do not agree with the expected proportions_ Calculate the test statistic, X2 , 72 = (Round to three decimal places as needed:_



Answers

Conduct the hypothesis test and provide the test statistic and the P-value and/or critical value, and state the conclusion. The table below lists the numbers of games played in 105 Major League Baseball (MLB) World Series. This table also includes the expected proportions for the numbers of games in a World Series, assuming that in each series, both teams have about the same chance of winning. Use a 0.05 significance level to test the claim that the actual numbers of games fit the distribution indicated by the expected proportions. $$\begin{array}{l|c|c|c|c} \hline \text { Games Played } & 4 & 5 & 6 & 7 \\ \hline \text { World Series Contests } & 21 & 23 & 23 & 38 \\ \hline \text { Expected Proportion } & 2 / 16 & 4 / 16 & 5 / 16 & 5 / 16 \\ \hline \end{array}$$

When has probably been given a probability distribution representing a number of famous played in the World Series from 1903 to 2000 and 12 You want to find the mean variance and standard deviation going to find the mean, which is the same as the expected value decision is equal to the sum of X times probability of acts. And so this is equal four times 0.176 plus five I was 0.241 pursuit times 0.213 lost seven times 0.333 plus eight times 37 When we evaluate this, this gives us our mean Okay, this 5.814 Now for the variants, this is equal to P of X squared minus e events squared. Now, First we find f X squared. Okay, this is equal to four squared times. The probability of 4.176 I mean was five squared times. The probability of 5.241 was six word times probability of six 1213 plus seven sward times the probability of seven, which is 70.333 plus eight squared times. The probability of eight. That's when we add all these together. Gives us 35.194 so he have expired is 35.194 and someone skiing are variants is equal to 35.194 that sea of X squared minus our average of just five point 814 square. This gives us 1.391 and so are variances, 1.391 Our standard deviation is the square root of the variance, or root of 1.391 which is one point 180 So that is our standard deviation on DSO to interpret the results using the context of the problem, The number of games played is on average, 5.814 which varies on average by 1.180 game mhm.

This question covers means and standard deviations of random um Random variables in a context, in this case it's Major League Baseball and how to apply that. Um In terms of calculating the population um mean and standard deviation. So we're given that there's two types of arab bolsters X and why, and I'd like to think of X as the population variables says population related and um why as a random variable? So like random sampling. Yeah, so for the first part were first asked to find what the mean of the random variable um is and what its standard deviation is. To recall that for a mean, the formula is some of the respective values times the probability. So the probability in this case the probability is the relative frequency. Um You notice that the add up to one, so really relative frequency is just equivalent to probability. Um Then we use that information to find the standard deviation, which is the square roots of the variable squared times the probability minus the mean squared. So yeah, let's get started. Um mu of y Mhm is equal to four times points to Right, four times point to plus five times 0.23 plus six times 0.22 plus seven times 0.35 Okay. Right. And our value that we should get Um is 5.72. Yeah, And since we're talking about a number of games here, it's 5.72 games. Don't forget to include what's your variable is then from this we can find the standard deviation of the variable which is the square roots four squared times about 25 squared times 0.23 plus six squared, that's 60.22 plus seven squared Times .35 minus the music. Where Um you that we got 5.72 squared. And what we should get from here is about 1.14 um 09 games Dan. for part two be. We want to consider um what how to find uh X verbals, given why we know that, we know that. Why is the mean of the mean and standard deviation of um random variable? But what is X, X. Is the population right? And were given a random sample, But if we take sufficient, why are mean is going to be equal to the population? And so is the standard deviation right? We take a sufficiently large sample, our distribution is going to look close to the population. So in that in that case we should expect that for X or the population, the total number of games in the world series to equal to why, Which is 5.7 two games, right? We should also expect that the standard deviation is also the same, so it's also one point um 1409 games. Yeah, you should we should expect them to be the same because um the random sample uh we're taking large enough samples, it approximates the population. We should expect them to be equivalent.

In a problem. Eight. We're going to be comparing the number of games one by each league. We have two legs here the American League and the National League for Baseball games for the years 1970 to 1993 at the 0.5 level of significance, We're going to see whether there is sufficient evidence to conclude that with a difference in the number of kings. So the first step is to state the non Harper. This is then get the critical values wrong. The data then one called the values that to make a decision, whether to reject or real, to reject the night policies. So in our case, the null hypothesis change not is that there is no difference in the number wins and the alternative put. This is a case that there is a difference in the number of wins and both cases comparing the theme to leaks. But next we just did the critical Bali Andi, In our case, the critical value at 0.5 level of significance is plus or minus 1.96 Next, we're going to give the ranking full each a win, and he had the rankings for the American League and for the National League. So after getting the ranking, we need to find out the sample size of a small A sample. And in this case, the smaller sample is for the national legs. Onda. We have 11 wins and 11 samples and then for the American League, we have 12. So we're going to get Are we going to sound the rungs for the National League? So it's going to be 2.5 sound all the way out to 22.5 and the sum is 125. Next, we get Newmar from substituting the values off anyone and entered into the formula. And that yields 132. You are is obtained by substituting evolution to this formula, and we get 16.24 the value of Zell calculated values that is going to be obtained from the substitution of the values that we have just obtained. That is 120 five minus 132 divided by 16.24 And when you want that, don't get negative 0.43 08 now we can compare that one, but you off that to the critical value offset and the negative. Negative. 0.43 08 He is great. German negative. 1.96 If you can sketch the distribution, you find that the critical value is negative. One 1.96 And this is the critical region. While I watch critical, our calculated value is zero negative 0.4. So negative 0.4308 does not fall in the critical region and hens we make the decision to feel to reject. Then my life with this is And in this case since we've failed to reject the knowledge by this is we can't conclude that these not enough evidence to reject the claim that there is no difference in the number winds in the tunings.

The following is a solution to number 23. And this looks at the number of games played in each World Series from 1923 2000 and five. So four games that happened. Uh So the World Series got done in four games that happened for 16 World Series, five games was 15 World Series, six games with 18 World Series. And then obviously they usually win seven games. That's 33 World Series and we're asked to find the probability distribution or created probability distribution. So we need the total number first. So if we were to add these together these frequencies 16 plus 15 plus 18 plus 33 We get 82 years worth of world series. So then we just take the frequency divided by the 82 to get the probability. And whenever we do that, draw a line here, we get up .195. So that's 16 divided by 82, 15 divided by 82 is .183 And then .22 and then point 40 to another. Good way to double check is so this is your probability distribution. These should add up pretty close to one. Now that maybe maybe a little off due to rounding but mine was one. Okay, and then part B is to make a history graham with those probabilities. So I kinda already set it up here. So the four was about point to close enough to point to maybe just a tick under. Okay, and then point 183. So just a little bit under that for the five And then the six was a little higher .22 And then seven was way up there, .33, I'm sorry, .40 .4. So I don't know where I got 33, maybe the 33, but yeah, that's what it is. Okay. And then the mean is the summation of X times p of x. So we would just take four times 40.195 Plus five times .183 Plus six times .22 Plus seven times .402. And that should give you five point 8 to 9. And also to interpret this. So the interpretation here is the average number of games played at all the World Series from 1923 to 2000 and five is 5.8 to 9 games. So it's the means, the average is the average number of the games played at all of the World Series in that time span. And the standard deviation is the square root of the variants. And the way to find the variance is you just take the X squared times the associated probability. So four square times 40.5, then we add that to five squared times that probability and then plus six squared times that probability And plus seven squared times the other probability. And then we subtract the mean squared and that gives you 1.3358. So we square root that to get one point 558, I'm Sorry, So that is the standard deviation in this problem.


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