Chapter 21, Problem 29E

(a)

To determine

To explain should the professor spend the money for this software.

Answer to Problem 29E

The professor should append the money for the software.

Explanation of Solution

It is given in the question that initially $203$ students signed up for the stats course. A statistics professor has observed that for several years about $13\%$ of the students who initially enroll in his introductory statistics course withdraw before the end of the semester. We can say that the null hypothesis states that the rate of students be equal to $13\%$ and the alternative hypothesis states that the rate of students be less than $13\%$ . So, we define hypothesis by:

  $\begin{array}{l}{H}_0:p=0.13\\ H_a:p<0.13\end{array}$

Thus, let us check the appropriate conditions and assumptions for inference to proceed further to test hypothesis as:

Random condition: It is satisfied as we assumed that the sample was randomly selected from the population.

Independent condition: It is satisfied as we can assume that the sample is independent.

  $10\%$ condition: It is satisfied as the sample is less than $10\%$ of all population.

Success/failure condition: It is satisfied because both are greater than ten as,

  $\begin{array}{l}np=11>10\\ n(1-p)=192>10\end{array}$

Thus, all the conditions are satisfied. Thus, for this first we have to select $5:1-\text{PropZInt}$ from the STAT TESTS menu. Then, enter the observed count and sample size for the sample. Then we will indicate what kind of test we want: one-tail lower tail, two-tail, or one-tail upper tail. Then, we will specify whether to calculate the result or draw the result. Thus, we have,

  $\begin{array}{l}x=11\\ n=203\\ c=0.95=95\%\end{array}$

So, by putting the values above by using the calculator we will get the result as,

The value of z-statistics and the P-value will be then,

  $\begin{array}{l}z=-3.21\\ P-\text{value}=0.0007\end{array}$

And as we know if the P-value is smaller than the significance level, then reject the null hypothesis, so, we have,

  $P<0.05\Rightarrow \text{ Reject }{H}_0$

Thus, we conclude that the dropout rate for the course is less than $13\%$ . So, the professor should append the money for the software.

(b)

To determine

To explain what your P-value means in this context.

Explanation of Solution

It is given in the question that initially $203$ students signed up for the stats course. a statistics professor has observed that for several years about $13\%$ of the students who initially enroll in his introductory statistics course withdraw before the end of the semester.

  $\begin{array}{l}{H}_0:p=0.13\\ H_a:p<0.13\end{array}$

The value of z-statistics and the P-value will be then,

  $\begin{array}{l}z=-3.21\\ P-\text{value}=0.0007\end{array}$

Thus, the P-value in this context means that the probability of observing $11$ or fewer dropouts in a class of $203$ is only $0.07\%$ of the dropout rate really is $13\%$ .