Question 11.3: Firearm Deaths A researcher read that firearm-related deaths......
Firearm Deaths
A researcher read that firearm-related deaths for people aged 1 to 18 years were distributed as follows: 74 \% were accidental, 16 \% were homicides, and 10 \% were suicides. In her district, there were 68 accidental deaths, 27 homicides, and 5 suicides during the past year. At \alpha=0.10, test the claim that the percentages are equal.
Source: Centers for Disease Control and Prevention.
Step 1 State the hypotheses and identify the claim:
H_{0} : The deaths due to firearms for people aged 1 through 18 years are distributed as follows: 74 \% accidental, 16 \% homicides, and 10 \% suicides (claim).
H_{1} : The distribution is not the same as stated in the null hypothesis.
Step 2 Find the critical value. Since \alpha=0.10 and the degrees of freedom are 3-1=2, the critical value is 4.605.
Step 3 Compute the test value.
First calculate the expected values, using the formula E=n \cdot p as shown.
\begin{aligned}& 100 \times 0.74=74 \\& 100 \times 0.16=16 \\& 100 \times 0.10=10\end{aligned}
The table looks like this.
\begin{array}{|l|c|c|c|} \hline \text{Frequency }& \text{Accidental }& \text{Homicides }& \text{Suicides }\\ \hline \text{Observed }& 68 & 27 & 5 \\ \text{Expected }& 74 & 16 & 10 \\ \hline \end{array}
\begin{aligned}\chi^{2} & =\sum \frac{(O-E)^{2}}{E} \\& =\frac{(68-74)^{2}}{74}+\frac{(27-16)^{2}}{16}+\frac{(5-10)^{2}}{10} \\& =10.549\end{aligned}
Step 4 Reject the null hypothesis, since 10.549>4.605, as shown in Figure 11-5.
Step 5 Summarize the results. There is enough evidence to reject the claim that the distribution is 74 \% accidental, 16 \% homicides, and 10 \% suicides.
