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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 null hypothesis can be stated as:

\(H_{0}:\) The number of regular applications accepted follows the same distribution as the number of early applications accepted.

And the alternative hypothesis can be stated as:

\(H_{a}:\) The number of regular applications accepted does not follow the same distribution as the number of early applications accepted.

\(df=5\)

\(p-value<\alpha\)

Therefore the decision will be to reject the null hypothesis, \(H_{0}\). Therefore, the null hypothesis gets rejected while alternative is accepted. Hence, it can be concluded that there are sufficient evidence to ensure that the number of regular applications accepted does not follow the same distribution as the number of early applications accepted.

Step by step solution

01

Step 1. Given information

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

Application Type Accepted

Brown

Columbia

Cornell

Dartmouth

Penn

Yale

Regular

2115

1792

5306

1734

2685

1245

Early Decision

577

627

1228

444

1195

761

02

Step 2. Calculation

The null hypothesis can be stated as:

\(H_{0}:\) The number of regular applications accepted follows the same distribution as the number of early applications accepted.

And the alternative hypothesis can be stated as:

\(H_{a}:\) The number of regular applications accepted does not follow the same distribution as the number of early applications accepted.

The degrees of freedom can be calculated by the formula given below;

\(df=\)(number of columns \(-1\)

\(\Rightarrow df=(6-1)\)

\(\Rightarrow df=5)\)

The observed value table is already given. Now let’s calculate the expected frequencies by using the formula shown below;

\(E=\frac{(row total)(column total)}{overall total}\)

Let’s calculate the expected (E) values in excel as shown below;

The test statistic of independence test is given below;

Test statistics \(=\sum_{i\times j}=\frac{(O-E)^{2}}{E}\)

To calculate \(\frac{(O-E)^{2}}{E}\) apply formula \(=((B4-B11)^{2})/B11\) in cell \(B17\) and drag the same formula up to cell F18. After that, take total of columns total and row total. The table of test statistic is shown below;

Hence, the test statistic is calculated as \(430.063\)

The p-value can be calculated in excel by using formula as shown below:

Therefore the p-value is zero.

The p-value sketch is shown below;

We have the alpha value given as \(0.05\). Now, since \(p-value<\alpha\) therefore the decision will be to reject the null hypothesis, \(H_{0}\) . Therefore, the null hypothesis gets rejected while alternative is accepted. Hence, it can be concluded that there are sufficient evidence to ensure that the number of regular applications accepted does not follow the same distribution as the number of early applications accepted.

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Most popular questions from this chapter

Suppose an airline claims that its flights are consistently on time with an average delay of at most 15 minutes. It claims that the average delay is so consistent that the variance is no more than 150 minutes. Doubting the consistency part of the claim, a disgruntled traveler calculates the delays for his next 25 flights. The average delay for those 25 flights is 22 minutes with a standard deviation of 15 minutes.

How did you know to test the variance instead of the mean?

The owner of a baseball team is interested in the relationship between player salaries and team winning percentage. He takes a random sample of 100 players from different organizations.

Use the following information to answer the next nine exercise: The following data are real. the cumulative number of AIDS cases reported for Santa Clary Country is broken down by ethnicity as in table 11.29

The percentage of each ethnic group in Santa Clary Country is as in Table shown below

If the ethnicities of AIDS victims followed the ethnicities of the total country population, fill in the expected number of cases per ethnic group.

Perform a goodness-of-fit test to determine whether the occurance of AIDS cases follows the ethnicities of the general population of Santa Clary Country

A sample of 212commercial businesses was surveyed for recycling one commodity; a commodity here means any one type of recyclable material such as plastic or aluminum. Table 11.41shows the business categories in the survey, the sample size of each category, and the number of businesses in each category that recycle one commodity. Based on the study, on average half of the businesses were expected to be recycling one commodity. As a result, the last column shows the expected number of businesses in each category that recycle one commodity. At the 5% significance level, perform a hypothesis test to determine if the observed number of businesses that recycle one commodity follows the uniform distribution of the expected values.

Business
Type
Number in
class
Observed Number that recycles one commodityExpected number that recycles one commodity
office35
19
17.5
Retail/
Wholesale
48
27
24
Food/
Restaurants
53
35
26.5
Manufacturing/
Medical
52
21
26
Hotel/Mixed24
9
12

Table 11.41

What are the null and alternative hypotheses for a math teacher who sees if two of her classes have the same distribution of test scores.

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