Question

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

The alpha value has been set at $0.05$. Because $p-value<\alpha$ , the null hypothesis, $H_0$, 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

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.

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.

$\begin{array}{|lllllll|}\hline \text{Application Type Accepted}& \text{Brown}& \text{Columbia}& \text{Cornell}& \text{Dartmouth}& \text{Penn}& \text{Yale}\\ \text{Regular}& 2115& 1792& 5306& 1734& 2685& 1245\\ \text{Early Decision}& 577& 627& 1228& 444& 1195& 761\\ \hline\end{array}$

The null hypothesis is as follows:

$H_0$: 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:

$H_a$: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.

$df=(\text{number of columns}-1)$

$\Rightarrow df=(6-1)$

$\Rightarrow df=5$

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

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

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

The independence test statistic 

The independence test statistic is as follows:

Statistics for the test $=\sum _{i\times j}=\frac{(O-E{)}^2}{E}$

Apply the formula $=\left((B4-B11{)}^2\right)/B11$to cell $B17$and drag the same formula up to cell $F18$to get $\frac{(O-E{)}^2}{E}$. 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;