Are the differences observed between the average hourly earnings of male and female workers statistically and significantly different from zero

INSTRUCTIONS TO CANDIDATES

ANSWER ALL QUESTIONS

Assignment

PART 1:

Multiple-choice questions:!! Canvas reshuffles the answers!!

(Q1) Which of the following statements is correct?

(a) R2 = 1 − (ESS/TSS).

(b) ESS = SSR + TSS.

(c) TSS = ESS + SSR.

(d) ESS > TSS.

At a recent county fair, you observed that at one stand people’s weight was forecasted, and were surprised by the accuracy (within a range). Thinking about how the person could have predicted your weight fairly accurately, you think about how this could have been accomplished. You remember that medical charts for children contain 5%, 25%, 50%, 75%, and 95% lines for a weight/height relationship and decide to conduct an experiment with 110 of your peers. You collect the data and calculate the following quantities:

(Q2) Compute the slope and intercept of the regression. (Q3) Interpret the slope.

Math SAT scores (Y ) are normally distributed with a mean of 500 and a standard devia- tion of 100. An evening school advertises that it can improve students’ scores by roughly a third of a standard deviation, or 30 points, if they attend a course which runs over several weeks. The statistician for a consumer protection agency suspects that the courses are not effective. She views the situation as follows: H0 : E(Y ) = 500 vs. H1 : E(Y ) = 530. The consumer protection agency wants to evaluate this claim by sending 50 students to attend classes. One of the students becomes sick during the course and drops out.

(Q4) What is the distribution of the average score of the remaining 49 students under the null, and under the alternative hypothesis?

(Q5) Compute the power of the test.

(Q6) What options does the statistician have for increasing the power in this situation without changing its size?

 PART 2:

This is a follow-up to Assignment #2 where we compared the population of male and female workers in 2008. From Assignment #2, we learnt that there were some differences in the distribution of average hourly earnings observed for male and female workers; more specifically, that female workers were on average earning less than male workers - and that these differences could not be explained by either education of age. We now want to test whether these differences are statistically and significantly different from zero.

1. Are the differences observed between the average hourly earnings of male and female workers statistically and significantly different from zero? Implement the appropriate test to answer this question using a significance level of 5%. We focus here on the whole population of workers.

(a) Compute the p-value of the test.

(b) What is the decision of the test?

(c) What is the critical assumption needed to implement your test?

2. Compute the power of the test for the following 11 alternative hypotheses: H1 : E(D) = x with x taking values 0, 0.1, 0.2, • • • , 1. Note: depending on how you write your null and alternative hypothesis, the relevant x value may be 0, −0.1, • • • , −1.

3. Represent the associated power curve.

For additional tips on how to compute these powers and the power curve, refer to the R code and associated PDF called Rchap4.pdf (available on canvas on the course roadmap for week 7).

The following R commands may be useful.

- To return the number of observations in a dataset, use nrow. Example: nrow(mydata)

- To print either the value of a variable, or a text, use print. For text statement, you need to use " " as in the example below.

Example:

print("The sky is blue") c=2

print(c)

- To do an action conditional on a statement, use if/else. The brackets { and } are critical!

Example:

if (c>0)

{print(c)} else

{print("c is negative")}

- Instead of using the Normal table, probabilities can be directly computed in R using the command pnorm. For example, to compute the probability that the standard normal is smaller than the number z, that is P ( (0, 1) < z), you can use: pnorm(z)

Example:

pnorm(1.96) [1] 0.9750021

Attachments:

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