Question

(Use the 6 point process as it concerns the determination of random variables and their distribution. Include the experiment(s), the form of the sample space(s), and write all probabilities in terms of appropriate and clearly defined random variables.) A proportion of people who have had a vaccine is 0.3 . It is expected that the proportion of a certain company's employees have a lower rate.

a. State the null and alternative hypotheses.
b. To test this it is suggested that you take a random sample of 10 employees and reject the null hypothesis if there are fewer than 2 reporting to have had the vaccine. What is the test size (level of significance α )?
c. Suppose instead it is suggested that you ask random employees if they have been vaccinated and you stop when the first says they have and you reject the null hypothesis if you have to ask 10 or more people before you find the first vaccinated employee. What is the test size then?
d. What is the impact of choosing one decision rule over the other?
e. Calculate the power for each decision rule if the true proportion is 0.2 .
f. Does the result in part e) change your thoughts about part d)? Explain.

Answer 1

The null and alternative hypotheses for the study are stated, the test size is calculated for two different decision rules, and the impact of choosing one decision rule over the other is explained. The power for each decision rule is calculated, and the potential impact of the power on the choice of decision rules is discussed.

Step-by-step explanation:

Question a:

State the null and alternative hypotheses.

To test if the proportion of the company's employees who have a lower rate of vaccination compared to the general population, we can state the null and alternative hypotheses as follows:

Null Hypothesis (H0): The proportion of the company's employees who have a lower rate of vaccination is equal to the general population proportion.
Alternative Hypothesis (Ha): The proportion of the company's employees who have a lower rate of vaccination is less than the general population proportion.

Question b:

To find the test size (level of significance α), we need to determine the critical value for the given test. Since we are testing if there are fewer than 2 employees reporting to have had the vaccine, which is a one-sided test, we can use the binomial distribution to find the critical value. At the 5% level of significance, we can calculate the critical value using the binomial distribution with parameters n = 10 (sample size) and p = 0.3 (general population proportion). The critical value will be the largest k such that P(X <= k) <= 0.05, where X is a binomial random variable representing the number of employees reporting to have had the vaccine in the sample. We can find this critical value using a binomial probability table or statistical software.

Question c:

To determine the test size if we stop when the first vaccinated employee is found, we need to calculate the probability of stopping at each possible number of employees questioned until the first vaccinated employee is found. Let X be a geometric random variable representing the number of employees questioned until the first vaccinated employee is found. Then, P(X = k) = (1-p)^(k-1) * p, where p is the proportion of individuals who have had the vaccine. The test size will be the sum of these probabilities when k >= 10.

Question d:

The impact of choosing one decision rule over the other is that it affects the type of error that could be made. The decision rule where we reject the null hypothesis if there are fewer than 2 employees reporting to have had the vaccine is a one-sided test and focuses on the possibility of missing employees who have had the vaccine. On the other hand, the decision rule where we reject the null hypothesis if we have to ask 10 or more people before finding the first vaccinated employee is also a one-sided test but focuses on questioning a large number of individuals before finding any vaccinated employee. The choice of decision rule depends on the specific research question and the priorities of the study.

Question e:

To calculate the power for each decision rule, we need to know the true proportion of the company's employees who have a lower rate of vaccination. Let p_true be the true proportion. The power of a hypothesis test is the probability of correctly rejecting the null hypothesis when it is false. For the decision rule of having fewer than 2 employees reporting to have had the vaccine, the power can be calculated as the probability of observing fewer than 2 successes in a binomial distribution with parameters n = 10 (sample size) and p = p_true. For the decision rule of having to ask 10 or more people before finding the first vaccinated employee, the power can be calculated as 1 minus the cumulative distribution function of a geometric distribution with parameter p = p_true evaluated at 9 (since the first vaccinated employee cannot be the 10th person). The power can be calculated using statistical software or online calculators.

Question f:

The result in part e) may change our thoughts about part d) depending on the values of p_true and the power for each decision rule. If the power of the decision rule with fewer than 2 employees reporting to have had the vaccine is higher than the power of the decision rule with having to ask 10 or more people before finding the first vaccinated employee for the true proportion p_true, we may prefer the decision rule of fewer than 2 employees. However, if the power of the decision rule with having to ask 10 or more people before finding the first vaccinated employee is higher, we may prefer that decision rule. The choice of decision rule ultimately depends on the trade-offs between the risk of missing vaccinated employees and the risk of questioning a large number of individuals without finding any vaccinated employee.