Question

Stating hypotheses State the appropriate null and alternative hypotheses in each of the following cases.

(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 (over $3000$students) differs from the national average. You measure an SRS of $48$female graduates and find that $\overline{X}=63.1$inches.

(b) Mr. Starnes believes that less than $75\%$of the students at his school completed their math homework last night. The math teachers inspect the homework assignments from a random sample of students at the school to help Mr. Starnes test his claim.

- Check conditions for carrying out a test about a population proportion or mean.

- Interpret $P$-values in context.

Short answer

(a) $H_0:\mu =64.2$

$H_{\alpha }\ne 64.2$

(b) $H_0:P=0.75$

$H_{\alpha }:P<0.75$

Step-by-step solution

Part (a)Step 1 :Given information

Given in the question that,

Population mean$\left(\mu \right)=64.2$

Sample mean$\left(\overline{x}\right)=63.1$

Sample size$\left(n\right)=48$we have to find an SRS of $48$female graduates and find that $\overline{X}=63.1$inches.

Part (a) Step2 :Explanation 

Consider $\mu$be the average height of $18$year old american women. using the information, the null and alternative hypothesis are:

$H_0:\mu =64.2$

$H_{\alpha }:\mu \ne 64.2$

Part (b) Step 1:Given Information

Given in the question that, population proportion $=0.75$

Part (b) Step 2:Explanation

Here, $p$be the population proportion of the students that completed their homework last night.

Using the information, the null and alternative hypotheses are:

$H_{\alpha }:P=0.75$

$H_{\alpha }:P<0.75$