Problem

M\&MMARS, makers of M\&M Chocolate Candies, condu…

06:31

Problem 13 Medium Difficulty

During the first 13 weeks of the television season, the Saturday evening $8 : 00$ p.M. to
$9 : 00$ p.M. audience proportions were recorded as ABC $29 \%,$ CBS $28 \%,$ NBC $25 \%,$ and
independents 18$\%$ . A sample of 300 homes two weeks after a Saturday night schedule
revision yielded the following viewing audience data: $A B C 95$ homes, CBS 70 homes,
NBC 89 homes, and independents 46 homes. Test with $\alpha=.05$ to determine whether the
viewing audience proportions changed.

Answer

There is not sufficient evidence to reject the claim of the specific distribution.

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Essentials of Modern Business Statistics

Chapter 11

Comparisons Involving Proportions and a Test of Independence

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Video Transcript

Okay. The problem is giving us for proportions. The shares of the viewership for ABC, CBS, NBC and the rest are called Independence. So we're gonna call that my Andy and the proportions of your shape, our 29% 28 25 and the remaining upto 100% that is 18%. This are the distributions that, um, will be expected to be mother. Now the problem there's us that the there's some revision in the schedule and then they study the study a group of 300 the viewers to not this to see if there is a change in these proportions the 300 to, ah, viewers. We're gonna we will call them the sample size and the results of the sample size are we have, um, that the results of the sample size will be called the observed frequency. So let's call them observed and the counts the observed frequency our 95 off the 300. By the way, these numbers here are percents or less put percent. Sign in the second column. These observed frequencies. 95 viewers. Then we have 70 viewers that watched CBS 89 viewers watched NBC and 46 viewers watched the 46 yours watched the independent channels. The problem is asking us to test if there's a difference in proportions. We have the observed values and now we need in order to the test, we need the expected frequencies or they expected counts. Um, in order to do that, because we have 300 we have this proportions, we need to calculate what would be the expected number off out of the 300 that, um, watch ABC. What would be the expected number for CBS s own? So if we think about it, the expected for ABC would be 29% off 300. Mathematically, we can calculate that by writing 29 as a decimal 290.29 off 300. And if you do the calculation, we were get an expected value off 87 yours. And if we continue that with same calculation for CBS NBC, an independent, the values will be 84 75 and the 54. So we notice some differences between observed frequencies and expected frequencies. But we want to measure somehow the magnitude of these differences. And to do that, we need toe uh, calculate a test statistic called Chi Square. So the chi Square test statistic the symbol? The Greek letter Kai. It's calculated by adding together, um set off chi square components, and each component is obtained by finding the absurd. Taking the observed observed value, absurd value in the formula is the notation is f sub I. Then we subtract the expected value. A sub I they expected is this column and the observed is this column. We have sometimes negatives and sometimes positive differences. So for that reason, way we square this difference and then we divided by the expected value. And if we add all those components, we will obtain Guy Square. Now, before we calculate that, I want to make sure we see the hypotheses off the test that we are performing and ah, the no hypotheses. Remember, there's two hypotheses that we need to check the North and the alternative. The no hypotheses will say that the through proportion of your shape that chooses ABC is 29% and sound. So we're used. In addition, p sub a. That's called it peace are Baby Sea. We have prophesized that the taste 29% the proportion for CBS 28. Proportioned for NBC is Ah, 25 then the last proportion. The proportion for independent is 18% and we want toe. See if we have evidence for the alternative hypotheses, which will say that the same proportions are not equal to those values. So the proportion of yourself for ABC is not equal to 0.29. It's on. The proportion for CBS is not equal to 28. The proportion for NBC is not equal to 25 then the last one is not equal to 80. And we're going to do that. The problem. There's us to check that at the artful ever significance level off 5%. So let's continue with the calculations will go back to our numbers. We have the expected. We have the observed. We need to calculate the difference. So that's right here. Half I minus E I. The observed frequency miners the expected frequency. So if you do the subtractions, we're going to have 95 minus 87 is a and then the next difference is negative. 14. Deny my positive 14 and then negative eight. So, as I said, we have positive and negative values of So for that reason, the numerator we're have to be squared. So if I minus e I squared, it was square These numbers we are going to get the value 64 8 square a four day square is 1 96 1 96 and 64. Again, we follow the formula. We continue to follow this chi square formula So we have the squared values we need to divide them by Ah by the expected if count So we have this squared various f i mine us e i squared defied it by going to move the sample size. So the karate a divided by e i so not to forget the sample size was 300. So when we divide those numbers, if you use a calculator and the brown let's run to the nearest 100. The values will be 0.74. The next ah component of the chi square is 2.33 approximately 2.61 and the 1.19 and the chi square is the sum off all these values. So once we add them together ah, we'll get the chi square off was we add 6.87. So the sky square comes to be a 6.87 and ah, toe come out. With the answer to the test, we need to use a table off values to calculate what is the area to the right, Um, under the chi square curve and the if you use that table, you will conclude that the P value for our problem is somewhere between, ah, 5% that 10%. So, um, we don't have the precise value, but we have an interval. It's bigger than 5%. In other words, it is bigger than the outfall ever bigger than the significance leg and the deaths of significance. You know, once we conclude that, then we can conclude that, um, we have toe fail to reject. The null hypothesis is when you fail to reject the null hypothesis is your conclusion is that you do not have evidence for the opportunity if you do not have evidence that the reason change in ah in, um, proportions of view off your ship. So to conclude, we will write, um, the conclusion scenes, people, Are you greater than are far, which was 5% we fare to reject the null hypothesis. Make sure in the second part being clue conclusion. We include context in the conclusion. So we're saying we do not have evidence that the viewership proportions changed. You could say change significantly.

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