Question

Stating hypotheses State the appropriate null and alternative hypotheses in each of the following settings. Explain why the sample data give some evidence for $H_a$ in each case.

a. The average height of $18-$year-old American women is $64.2$ inches. You wonder whether the mean height of this year’s female graduates from a large local high school differs from the national average. You measure an SRS of $48$ female graduates and find that $\overline{x}=63.5$ inches and $s_x=3.7$ inches.

b. Rob once read that one-quarter of all people have played/danced in the rain at some point in their lives. His friend Justin thinks that the proportion is higher than $0.25$ for their high school. To settle their dispute, they ask a random sample of $80$ students in their school and find out that $28$ have played/danced in the rain.

Short answer

Part (a)$$\begin{gathered} H_0:\mu =64.2 \\ {H}_1:\mu \ne 64.2 \end{gathered}$$

Part (b) $$\begin{gathered} H_0:p=0.25 \\ {H}_1:p>0.25 \end{gathered}$$

Although the sample proportion is $0.35$ which is higher than $0.25$ the sample proportion is giving some evidence for $H_1$

Step-by-step solution

Part (a) Step 1: Given information

The claim is mean differs from $64.2$

Part (a) Step2: Calculation

The null hypothesis asserts that the population value should be the same as the claim value.

$H_0:\mu =64.2$

Either the null hypothesis or the alternative hypothesis is the assertion. The null hypothesis asserts that the population means should be the same as the claim value. If the claim is the null hypothesis, the alternative hypothesis statement is the polar opposite of the claim.

$H_1:\mu \ne 64.2$

Although the sample mean of $63.5$differs from the hypothesized mean of $64.2$ there is some indication of $H_1$ in the sample averages.

Part (b) Step 1: Calculation

The null hypothesis asserts that the population value should be the same as the claim value.

$H_0:p=0.25$

Either the null hypothesis or the alternative hypothesis is the assertion. The null hypothesis asserts that the population proportion should be the same as the claim value. If the claim is the null hypothesis, the alternative hypothesis statement is the polar opposite of the claim.

$H_1:p>0.25$

Here, $p$ is the percentage of pupils in that school who have played in the rain.

$$\begin{gathered} x=28 \\ n=80 \\ \hat{p}=\frac{x}{n}=\frac{28}{80} \\ =0.35 \end{gathered}$$

Despite the fact that the sample proportion is $0.35$which is greater than $0.25$the sample proportion supports $H_1$