# 9.1.2.1.2 – Example: Same Sex Marriage

## Example: Same Sex Marriage Section

A survey was given to a random sample of college students. They were asked whether they think same sex marriage should be legal. We'll compare the proportion of males and females who responded "yes." Of the 251 females in the sample, 185 said "yes." Of the 199 males in the sample, 107 said "yes."

1. Check any necessary assumptions and write null and alternative hypotheses.

For females, there were 185 who said "yes" and 66 who said "no." For males, there were 107 who said "yes" and 92 who said "no." There are at least 10 successes and failures in each group so the normal approximation method can be used.

• $$\widehat{p}_{f}=\frac{185}{251}$$
• $$\widehat{p}_{m}=\frac{107}{199}$$

This is a two-tailed test because we are looking for a difference between males and females, we were not given a specific direction.

• $$H_{0} : p_{f}- p_{m}=0$$
• $$H_{a} : p_{f}- p_{m}\neq 0$$
2. Calculate an appropriate test statistic.

$$\widehat{p}=\frac{185+107}{251+199}=\frac{292}{450}=0.6489$$

$$SE_0=\sqrt{\frac{292}{450}\left ( 1-\frac{292}{450} \right )\left ( \frac{1}{251}+\frac{1}{199} \right )}=0.0453$$

$$z=\frac{\frac{185}{251}-\frac{107}{199}}{0.0453}=4.400$$

Our test statistic is $$z=4.400$$

3. Determine a p value associated with the test statistic.

$$P(z>4.400)=0.0000054$$, this is a two-tailed test, so this value must be multiplied by two: $$0.0000054\times 2= 0.0000108$$

$$p<0.0001$$

4. Decide between the null and alternative hypotheses.

$$p\leq0.05$$, therefore we reject the null hypothesis.

5. State a "real world" conclusion.

There is evidence that there is a difference between the proportion of females and males who think that same sex marriage should be legal.