Question

Show all five steps: The harmful effects of smoking tobacco have been illustrated in medical studies yet many individuals still choose to smoke. The Centers for Disease Control estimate that approximately 13% of students aged 18 – 24 smoke tobacco. Using only the students in the “22 or younger” age group test to see if the proportion of students smoking tobacco is lower in Maryland. Let α= 0.01.  

Data from StatCrunch: 1,129 students aged 22 or younger surveyed 51 "yes", 1078 "no"

Please list each step: Hypotheses, Conditions/Assumptions, Mechanics, Decision, and Statement

Answer 1

Solution:

Given: . The Centers for Disease Control estimate that approximately 13% of students aged 18 – 24 smoke tobacco.

That is: p = proportion of students aged 18 – 24 smoke tobacco = 0.13

Sample size = n = Number of students aged 22 or younger = 1129

x = Number of  students aged 22 or younger smoke tobacco = 51

Level of significance = α = 0.01.  

We have to test the proportion of students smoking tobacco is lower in Maryland  in the “22 or younger” age group. Thus this directed to lower side, so this is left tailed test.

Thus we use following steps:

Step 1) State H0 and H1:

H0: p = 0.13 Vs H1: p < 0.13

Step 2) Conditions/Assumptions:

n * p = 1129 * 0.13 = 146.77 > 10 and n*(1-p) = 1129 * (1-0.13) =1129 * 0.87 = 982.23 > 10

Both conditions are satisfied. Thus we can approximate Normal distribution to proportion and use Normal distribution to test the hypothesis.

Step 3) Mechanics :

That is find test statistic and Critical value

Test statistic:

where

Thus

Critical value:

Level of significance = α = 0.01.  

look in z table for area = 0.0100 or its closest area and find z value.

Area 0.0099 is closest to 0.0100 and it corresponds to -2.3 and 0.03

thus z critical value = -2.33

Step 4) Decision:

Decision rule: reject null hypothesis H0, when z test statistic value < z critical value = -2.33, otherwise we fail to reject H0.

Since z test statistic value = < z critical value = -2.33, we reject null hypothesis H0.

Step 5) Statement of conclusion:

Since we have rejected null hypothesis H0, at 0.01 level of significance , there is sufficient evidence to conclude that: the proportion of students smoking tobacco is lower in Maryland