How well do airline companies serve their customers? A study showed the following
customer ratings: 3$\%$ excellent, 28$\%$ good, 45$\%$ fair, and 24$\%$ poor (BusinessWeek,
September $11,2000$ ). In a follow-up study of service by telephone companies, assume
that a sample of 400 adults found the following customer ratings: 24 excellent, 124 good,
172 fair, and 80 poor. Is the distribution of the customer ratings for telephone companies
different from the distribution of customer ratings for airline companies? Test with
$\alpha=.01 .$ What is your conclusion?
There is sufficient evidence to reject the claim of the specific distribution.
this problem were asked to test whether the distribution of ratings for a telephone company are different than the distribution of ratings for an airline company. So we're given that, um the probability people finding Ah, this excellent is 0.3 the proportion, not probability. So the proportion of people that find it good is equal to 0.28 proportion of people that find it fair is equal 0.45 and the fortune of people that find it poor is equal to 0.24 And now our alternative hypothesis is that not all of these are equal. Not oh, proportions O R equal. So this is not necessarily true and were given the following information that these are our expected proportions. Those are the customer ratings for telephone companies. And this is the, um, sorry. This is the ratings by airline companies, and these are the ratings for telephone companies and were given that we have we sampled for the telephone company's 400 people. So we sampled 400 people. And now we have to find, um, what would the expected frequency be for airline companies, given that we sampled 400 people so that would simply be the proportion times our end. So 0.3 times 0.4 times four hundreds are 24.28 times 400 is 1 24 Um, 0.45 times 400 is 1 72 and 0.24 times 100 is equal to E. Oh, I wrote down in the wrong values. Ah, sorry about that. So the expected frequency 0.3 times 400 is 12 0.28 times 400 is one 12 0.45 times 400 is 1 80 and 0.24 times 400 is equal to 96. So now we have to find, um, hour test statistic. And we're going to use a Chi Square, Tess statistic to find the goodness of fit. And this is the formula for sky squared goodness of fit test. We're looking for the difference between the actual frequency and the expected frequency squared, divided by the expected frequency for each category. Then we're going to take the sum of this value for all the categories. So given this, we have to find the difference. So we're going to take 24 minus 12 which is 12 1 24 minus 12 which is 12 1 72 minus 1 80 which is negative. Eight and 80 minus 96 which is negative. 16. And now we're going to square our differences. So we get 1 44 1 44 64 and one or 2 56 And now we have to find the difference squared over the expected frequency. Then we're going to take some of that to find our cart chi square test statistic. So this is equal to 12 is equal to 1.29 This is equal to 0.36 and this is equal to 2.67 And once we, and to find our chi squared, we're going to take some of these values. And some of these values is equal to approximately 16.3. So now, using our chi squared value of 16.3 and we have to find a degrees of freedom in order to compute a p value, our degrees of freedom is simply the number of categories minus one, and from here we have four categories, so that's four minus one, which is equal to three. So our p value is equal probability. Um, of arc I squared equals 16.3 with the degrees of freedom of three, which is equal to, um or which is less than 0.5 And now we're asked to come up with the conclusion, given that we haven't Alfa of 0.1 so and our P value is less than 0.5 So because our P value is less than 0.5 and 0.5 is less than 0.1 we reject the no. So what does that mean? It means that they're sufficient evidence to reject the claim that to the ratings for the airline and telephone companies are the same.