Question

99. A recent drug survey showed an increase in the use of drugs and alcohol among local high school seniors as compared to the national percent. Suppose that a survey of 100 local seniors and 100 national seniors is conducted to see if the proportion of drug and alcohol use is higher locally than nationally. Locally, 65 seniors reported using drugs or alcohol within the past month, while 60 national seniors reported using them.

Short answer

(a) The null hypothesis: $H_0:p_1\le p_2$
(b) The alternate hypothesis: $H_a:p_1>p_2$
(c) The random variable is the difference in the proportions of high school seniors who use alcohol and drugs in local and national high schools.
(d) For two proportions, normal.
(e) Test statistics $0.730$

(f) The $p$-value is $0.2326$.
(g) (i) $\alpha =0.05$
(ii) Decision: reject the null hypothesis.
(iii) Reason for Decision: $p-value>\alpha$.
(iv) As a result: There is insufficient information to establish that the proportion of local high school seniors who use drugs and alcohol is higher than the proportion of national high school seniors at the $5\%$significance level.

Step-by-step solution

Given information

Let, the survey of $100$ local seniors and $100$ national seniors is conducted to see if the proportion of drug and alcohol use is higher locally than nationally.

Explanation

(a) The null hypothesis is indicated as:
$H_0:p_1\le p_2$
(b) Alternate hypothesis is indicated as:
$H_a:p_1>p_2$
(c) The random variable is the difference in the proportions of high school seniors who use alcohol and drugs in local and national high schools.
(d) For two proportions, normal.
(e) To determine the test static as follows:
To access the stat list editor, click STAT and then 1.

Then ENTER all the values the OUTPUT will be:

Hence, the test statistics is $0.730$.

Explanation

(f) The $p$-value from the output is determined as $0.2326$.
(g) Obtain a clear picture of the situation using the information from the last task.
The horizontal axis should be clearly labelled and scaled, and the region(s) corresponding to the $p$-value should be shaded on the graph.

(i) $\alpha =0.05$
(ii) Decision: reject the null hypothesis.
(iii) Reason for Decision: $p-value>\alpha$.
(iv) As a result: There is insufficient information to establish that the proportion of local high school seniors who use drugs and alcohol is higher than the proportion of national high school seniors at the 5% significance level.