Question

Let us consider the following scenario:

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.
a. Let X1 be the number of local seniors who reported using drugs or alcohol withing the past month. What are the values of the random variable?
b. What is the distribution of X1?
c. What is the best point estimate of p1, the population proportion of local seniors who reported using drugs or alcohol withing the past month?
d. What is the best point estimate for p1 – p2, the difference in proportions for local and national?
e. What is the statement of the null hypothesis for this test?
f. What distribution would you use if you conducted the hypothesis test?
g. Locally, 65 seniors reported using drugs or alcohol within the past month, while 60 national seniors reported using them. Construct the 95% confidence interval for the difference in proportions and interpret the result.

Answer 1

The questions relate to statistical analysis of drug and alcohol use among local high school seniors versus the national percent, involving a random variable, distribution, point estimates, hypothesis testing, and construction of confidence intervals.

Understanding the Random Variable and Hypothesis Testing in Drug Survey Study

In the given scenario, where the drug and alcohol use among local high school seniors is compared to the national percent, several statistical concepts are explored:

• a. The random variable X1 represents the number of local seniors who reported using drugs or alcohol within the past month, and its values could range from 0 to 100.
• b. The distribution of X1 is binomial because it counts the number of successes (in this case, drug or alcohol use) in a fixed number of trials (100 seniors), with each senior representing a Bernoulli trial.
• c. The best point estimate of p1, the population proportion of local seniors who used drugs or alcohol, would be the sample proportion, calculated by X1/100.
• d. The best point estimate for p1 – p2, the difference between local and national proportions, is found by taking the difference between the two sample proportions.
• e. The null hypothesis (H0) for this test could state that there is no difference in the proportion of drug and alcohol use between local and national seniors, that is, p1 = p2.
• f. To conduct the hypothesis test, we would use the normal distribution for the difference in sample proportions, assuming that the sample size is large enough and the populations are independent.
• g. With 65 local seniors and 60 national seniors reporting usage, a 95% confidence interval for the difference in proportions can be constructed using the formula for two independent samples.
• This interval can then be interpreted to assess the estimate of the difference in proportions with a specified level of confidence.

In summary, understanding the distribution of the random variable, forming the correct null hypothesis, and using the appropriate distribution are critical steps in analyzing the survey's outcome and determining if the increase in drug and alcohol use is statistically significant.