Question

according to the centers of disease control, about 21% of american young people between the ages of 12 and 19 are obese.
a student claims that the students in his local schools are in better shape than the national average, so their proportion of obese students is less than 21%.
which are the appropriate null and alternative hypotheses for this study?
$\mathrm{h}_0: p = 21\\%, \mathrm{h}_a: p > 21\\%$
$\mathrm{h}_0: p = 21\\%, \mathrm{h}_a: p < 21\\%$
$\mathrm{h}_0: p \
eq 0, \mathrm{h}_a: p = 0$
$\mathrm{h}_0: p = 21\\%, \mathrm{h}_a: p \
eq 21\\%$
suppose that the result is to reject $\mathrm{h}_0$.
which is the proper conclusion?
there is not enough information to answer this question.
there is sufficient evidence to support the claim that the students at his local schools have a smaller proportion of obese students.
there is not sufficient evidence to support the claim that the students at his local schools have a smaller proportion of obese students.

Explanation:

Part 1: Null and Alternative Hypotheses

The null hypothesis (\(H_0\)) is a statement of no difference or the status quo. Here, the national average proportion of obese students is 21%, so \(H_0: p = 21\%\). The alternative hypothesis (\(H_A\)) reflects the student's claim that the local proportion is less than 21%, so \(H_A: p < 21\%\).

Rejecting the null hypothesis (\(H_0: p = 21\%\)) means there is sufficient evidence to support the alternative hypothesis (\(H_A: p < 21\%\)), which is the student's claim that the local school's proportion of obese students is smaller.

Answer:

\(H_0: p = 21\%, H_A: p < 21\%\) (the second option)

Part 2: Conclusion when \(H_0\) is Rejected