# 8.1.2.1.2 - Example: Handedness

Research Question: Are more than 80% of American's right handed?

In a sample of 100 Americans, 87 were right handed.

1. Check assumptions and write hypotheses

$$np_0 = 100(0.80)=80$$

$$n(1-p_0) = 100 (1-0.80) = 20$$

Both $$np_0$$ and $$n(1-p_0)$$ are at least 10 so we can use the normal approximation method.

This is a right-tailed test because we want to know if the proportion is greater than 0.80.

$$H_{0}\colon p=0.80$$
$$H_{a}\colon p>0.80$$

2. Calculate the test statistic
Test statistic: One Group Proportion

$$z=\dfrac{\widehat{p}- p_0 }{\sqrt{\frac{p_0 (1- p_0)}{n}}}$$

$$\widehat{p}$$ = sample proportion
$$p_{0}$$ = hypothesize population proportion
$$n$$ = sample size

$$\widehat{p}=\dfrac{87}{100}=0.87$$, $$p_{0}=0.80$$, $$n=100$$

$$z= \dfrac{\widehat{p}- p_0 }{\sqrt{\frac{p_0 (1- p_0)}{n}}}= \dfrac{0.87-0.80}{\sqrt{\frac{0.80 (1-0.80)}{100}}}=1.75$$

Our $$z$$ test statistic is 1.75.

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

This is a right-tailed test so we need to find the area to the right of the test statistic, $$z=1.75$$, on the z distribution.

Using Minitab Express, we find the probability $$P(z\geq1.75)=0.0400592$$ which may be rounded to $$p\; value=0.0401$$.

4. Make a decision

$$p\leq .05$$, therefore our decision is to reject the null hypothesis

5. State a "real world" conclusion

Yes, there is statistical evidence to state that more than 80% of all Americans are right handed.