Refer to the World Development (February 1998) study of street vendors’ earnings
ID: 3020049 • Letter: R
Question
Refer to the World Development (February 1998) study of street vendors’ earnings (y), Exercise 4.6 (p. 184). The SAS printout below shows both a 95% prediction interval for y and a 95% confidence interval for E(y) for a 45-year-old vendor who works 10 hours a day (i.e., for x1 = 45 and x2 = 10).
(a) Interpret the 95% prediction interval for y in the words of the problem.
(b) Interpret the 95% confidence interval for E(y) in the words of the problem.
(C) Note that the interval of part a is wider than the interval of part b. Will this always be true? Explain.
SAS output for Exercise 4.20 Dependent Variable: EARNINGS Output Statistice Dep Var Predicted Std Error Obe AGE HOURS EARNINGS 16 45 Value Mean Predict 95%CL Predict 10 3018 182.3519 1760 4275 Std Error Value Mean Predict Dep Var Predicted Obs AGE HOURS EARNINGS 95% CL Mean 182.3519 2620 3415 16 45 10 3018Explanation / Answer
Let dependent variable y : street vendors’ earnings
The SAS printout below shows both a 95% prediction interval for y and a 95% confidence interval for E(y) for a 45-year-old vendor who works 10 hours a day (i.e., for x1 = 45 and x2 = 10).
(a) Interpret the 95% prediction interval for y in the words of the problem.
We see in the SAS output that 95% prediction interval for y is (1760, 4275) and the predicted value is 3018.
From this information we see that predicted value lie in the confidence interval so accept null hypothesis.
(b) Interpret the 95% confidence interval for E(y) in the words of the problem.
The 95% confidence interval for E(y) is (2620, 3415) and predicted value for mean is 3018.
We can conclude that predicted value for mean also lie in the confidence interval so accept null hypothesis.
(C) Note that the interval of part a is wider than the interval of part b. Will this always be true? Explain.
length of first confidence interval = 4275 - 1760 = 2515
length of second confidence interval = 3415 - 2620 = 795
This is not always true sometimes the intervals are narrower.
Higher confidence (confidence level 90%, 95%, 99% etc) has a wider interval.
Smaller sample size would result in a wider confidence interval.