Question

1. The College Alcohol Study interviewed a sample of 2,100 college students about drinking habits and 1,519 supported cracking down on underage drinking. Is there enough evidence to suggest that less than 75% of all college student support cracking down on underage drinking?

Answer 1

Solution :

The null and alternative hypotheses are as follows :

H​​​​​​0 : p = 0.75 i.e. The proportion of all college student who support cracking down on underage drinking is 0.75 .

H​​​​​​1 : p < 0.75 i.e. The proportion of all college student who support cracking down on underage drinking is less than 0.75.

To test the hypothesis we shall use z-test for single proportion. The test statistic is given as follows :

Where, p̂ is sample proportion, p is hypothesized value of population proportion under H​​​​​​​​​​​0, q = 1 - p and n is sample size.

Sample proportion of college student who supported cracking down on underage drinking is,

p = 0.75, q = 1 - 0.75 = 0.25 and n = 2100

The value of the test statistic is -2.8221.

Since, our test is left-tailed test, therefore we shall obtain left-tailed p-value for the test statistic. The left-tailed p-value is given as follows :

p-value = P(Z < value of the test statistic)

p-value = P(Z < -2.8221)

p-value = 0.0024

The p-value is 0.0024.

We make decision rule as follows :

If p-value is less than the significance level, then we shall reject the null hypothesis (H​​​​​​0) at given significance level.

If p-value is greater than the significance level, then we shall be fail to reject the null hypothesis (H​​​​​​0) at given significance level.

In our question, significance level is not given. Generally significance level of 0.01 or 0.05 is used. We shall use significance level of 0.01.

Significance level (α) = 0.01

(0.0024 < 0.01)

Since, p-value is less than the significance level of 0.01, therefore we shall reject the null hypothesis (H​​​​​​​​​​​0) at 0.01 significance level.

Conclusion : At 0.01 significance level, there is sufficient evidence to conclude that the less than 75% of all college student support cracking down on underage drinking.

Please rate the answer. Thank you.