Question

1. Studies conducted in the 1970s indicated that the average age at which children take their first alcoholic drink is 14.6 years old. Sociologists believe that children are starting to drink at a younger age. A random sample of 169 young adults (18 years of age) is selected and the age at which each adult took their first alcoholic drink is recorded. The sample mean age was 13.3 years of age with a standard deviation of 5 years.

For the problems above perform a hypothesis test. State the null and alternative hypothesis, the p-value, and your conclusion in the context of the problem. also state if you used a t or z test. Perform the test at a .05 significance.

Answer 1

Let X denote the age of children when they first take their alcoholic drink

Null hypothesis,H​0 : = 14.6 years

Alternative hypothesis, H​​​​​​a : < 14.6 years

Sample mean, = 13.3 years

Standard deviation, = 5 years

Sample size, n = 169

As the sample size is 169 ( > 30) and the standard deviation is known, from the Central Limit Theorem the sampling distribution of the sample mean follows Normal distribution and hence z test can be used

z statistic = = = -3.38

Thus, the p-value = P(Z < -3.38) (Since it is a left tailed test)

= 0.0004

Since the p-value is less than the significance level, we reject the null hypothesis.

Thus, it can be concluded that the children are starting to drink at a younger age (age less than 14.6 years)