Question

1. a study was conducted at a police academy to analyze whether preference for tasers or pepper spray is related to the gender of the cadet. the table below shows the results. at \\( \alpha = 0.05 \\), test the claim that weapon preference and gender are independent.\

\

	female	male	\
---------------	--------	------	\
tasers	18	40	\
pepper spray	32	25	\

\
given: the test value is \\( \chi^2 = 7.373 \\).\
\
use the traditional method of hypothesis testing.\
(a) state the hypotheses and identify the claim.\
(b) find the critical value(s).\
(c) compare the test value to the critical value(s).\
(d) make the decision to reject or not reject the null hypothesis.\
(e) summarize the results.

Explanation:

Step1: State Hypotheses

• Null Hypothesis ($H_0$): Weapon preference and gender are independent.
• Alternative Hypothesis ($H_1$): Weapon preference and gender are dependent (claim).

Step2: Find Critical Value

• This is a chi - square test for independence. The contingency table has $r = 2$ (rows: Tasers, Pepper Spray) and $c=2$ (columns: Female, Male) categories.
• Degrees of freedom $df=(r - 1)(c - 1)=(2 - 1)(2 - 1)=1$.
• For $\alpha = 0.05$ and $df = 1$, from chi - square distribution table, critical value $\chi^2_{0.05,1}=3.841$.

Step3: Compare Test and Critical Values

• Test value $\chi^2=7.373$, critical value $\chi^2_{0.05,1}=3.841$.
• Since $7.373>3.841$, test value > critical value.

Step4: Make Decision

• Reject the null hypothesis $H_0$ because test value exceeds critical value.

Step5: Summarize Results

• There is sufficient evidence at $\alpha = 0.05$ to support the claim that weapon preference is related to the gender of the cadet.

Answer:

(a) $H_0$: Weapon preference and gender are independent; $H_1$: Weapon preference and gender are dependent (claim).
(b) Critical value: $3.841$
(c) $7.373>3.841$
(d) Reject $H_0$
(e) There is sufficient evidence to support the claim that weapon preference is related to gender.