Question

Steroids in high school A study by the National Athletic Trainers Association surveyed random samples of $1679$high school freshmen and 1366 high school seniors in Illinois. Results showed that $34$of the freshmen and $24$of the seniors had used anabolic steroids. Steroids, which are dangerous, are sometimes used in an attempt to improve athletic performance. Researchers want to know if there is a difference in the proportion of all Illinois high school freshmen and seniors who have used anabolic steroids.

a. State appropriate hypotheses for performing a significance test. Be sure to define the parameters of interest.

b. Check if the conditions for performing the test are met.

Short answer

a. The hypothesis are: $H_0:p_1=p_2$and $H_a:p_1\ne p_2$

b. All conditions are met.

Step-by-step solution

Given Information

It is given that researchers want to know if there is difference in proportion of all Illinois high school and seniors using steroids or not.

$x_1=34$

$x_2=24$

$n_1=1679$

$n_2=1366$

Hypothesis for performing significance test

Claim is difference between the proportions.

Appropriate hypothesis are:

Null: $H_0:p_1=p_2$

Alternative hypothesis: $H_a:p_1\ne p_2$

$p_1$isrole="math" localid="1654706370645" $\text{proportionfreshmen in Illinois that used anabolic steroids.}$

$p_2$is$\text{proportion ofseniors in Illinois that used anabolic steroids.}$

Checking the conditions

Conditions are:

Random: Samples are independent random samples from separate populations.

Independent: $1679<10\%$of high school freshmen and $1366<10\%$of high school seniors.

Normal: There are $34$success and $1679-34=1645$failures in first sample and $24$success and $1366-24=1342$failures in second sample, all are less than ten.

All conditions are satisfied, we can use hypothesis test.