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Ivy League schools receive many applications, but only some can be accepted. At the schools listed in Table 11.22, two types of applications are accepted: regular and early decision.


We want to know if the number of regular applications accepted follows the same distribution as the number of early applications accepted. State the null and alternative hypotheses, the degrees of freedom and the test statistic, sketch the graph of the p-value, and draw a conclusion about the test of homogeneity.

Short Answer

Expert verified

The alpha value has been set at 0.05. Because p-value<α , the null hypothesis, H0, will be rejected. As a result, the null hypothesis is rejected, whereas the alternative hypothesis is accepted. As a result, enough evidence exists to suggest that the number of regular applications approved does not follow the same distribution as the number of early applications accepted.

Step by step solution

01

Given information

Given in the question that,

Ivy League schools receive many applications, but only some can be accepted. At the schools listed in Table 11.22, two types of applications are accepted: regular and early decision.

We want to know if the number of regular applications accepted follows the same distribution as the number of early applications accepted. And also state the null and alternative hypotheses, the degrees of freedom and the test statistic, sketch the graph of the p-value, and draw a conclusion about the test of homogeneity.

02

Explanation

Ivy League colleges receive a large number of applications, yet only a small percentage of them are approved. Two types of applications are accepted at the schools listed in the table below: regular and early decision.

Application Type AcceptedBrownColumbiaCornellDartmouthPennYaleRegular211517925306173426851245Early Decision57762712284441195761

The null hypothesis is as follows:

H0: The number of regular applications approved is distributed in the same way that the number of early applications is distributed.

The alternative hypothesis is as follows:

Ha:The number of regular applications accepted is not distributed in the same way as the number of early applications accepted is distributed.

The formula below can be used to determine degrees of freedom.

df=(number of columns-1)

df=(6-1)

df=5

The table of observed values has already been provided. Let's now use the formula below to determine the predicted frequencies:

E=(row total)(column total)overall total

Let's use Excel to determine the expected (E) values, as shown below.

03

The independence test statistic 

The independence test statistic is as follows:

Statistics for the test =i×j=(O-E)2E

Apply the formula =(B4-B11)2/B11to cell B17and drag the same formula up to cell F18to get (O-E)2E. After that, add up the totals of the columns and rows. The following is a table of test statistics:

As a result, the test statistic is 430.063.

The p-value can be determined using the CHIDIST () function in Excel, as illustrated below:

Therefore the p-value is zero.

The p-value sketch is shown below;

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Most popular questions from this chapter

Do men and women select different breakfasts? The breakfasts ordered by randomly selected men and women at a popular breakfast place are shown in Table. Conduct a test for homogeneity at a 5%level of significance.

French ToastPancakesWafflesOmelettesMen47352853Women65595560

A fisherman is interested in whether the distribution of fish caught in Green Valley Lake is the same as the distribution of fish caught in Echo Lake. Of the 191 randomly selected fish caught in Green Valley Lake, 105 were rainbow trout, 27 were other trout, 35 were bass, and 24 were catfish. Of the 293 randomly selected fish caught in Echo Lake, 115 were rainbow trout, 58 were other trout, 67 were bass, and 53 were catfish. Perform a test for homogeneity at a 5% level of significance.

Determine the appropriate test to be used in the next three exercises.

An economist is deriving a model to predict outcomes on the stock market. He creates a list of expected points on the stock market index for the next two weeks. At the close of each day's trading, he records the actual points on the index. He wants to see how well his model matched what actually happened.

How many passengers are expected to travel between 401 and 500 miles and purchase first-class tickets?

A plant manager is concerned her equipment may need recalibrating. It seems that the actual weight of the 15oz. cereal boxes it fills have been fluctuating. The standard deviation should be at most 0.5oz. In order to determine if the machine needs to be recalibrated, 84randomly selected boxes of cereal from the next day’s production were weighed. The standard deviation of the 84 boxes was 0.54. Does the machine need to be recalibrated?

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