Chapter 20, Problem 31E

(a)

To determine

To explain is the death rate among the movie crew unusually high.

Answer to Problem 31E

The death rate among the movie crew is unusually high.

Explanation of Solution

It is given in the question that a total of $46\text{ of the 220}$ people working on the film eventually died of cancer. Thus, the null hypothesis states that the rate is equal to $13.6\%$ . And the alternative hypothesis states that the rate is more than $13.6\%$ . So, we define hypothesis as:

  $\begin{array}{l}{H}_0:p=0.136\\ H_a:p>0.136\end{array}$

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,

  $\begin{array}{l}np=0.136(220)>10\\ n(1-p)=0.864(420)>10\end{array}$

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,

  $\begin{array}{l}x=46\\ n=220\\ c=0.95=95\%\end{array}$

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,

  $\begin{array}{l}z=3.163\\ P-\text{value}\approx 0\end{array}$

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 death rate among the movie crew is unusually high.

(b)

To determine

To explain does this prove that exposure to radiation increases the risk of cancer.

Answer to Problem 31E

No, this does not prove that exposure to radiation increases the risk of cancer.

Explanation of Solution

It is given in the question that a total of $46\text{ of the 220}$ people working on the film eventually died of cancer. Thus, the null hypothesis states that the rate is equal to $13.6\%$ . And the alternative hypothesis states that the rate is more than $13.6\%$ . So, we define hypothesis as:

  $\begin{array}{l}{H}_0:p=0.136\\ H_a:p>0.136\end{array}$

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

  $\begin{array}{l}z=3.163\\ P-\text{value}\approx 0\end{array}$

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 death rate among the movie crew is unusually high. Thus, we can say that no, this does not prove that exposure to radiation increases the risk of cancer because there is no definitive proof. We cannot prove anything, only support our conclusion.