1. Statistical Inference: Confidence Intervals

File AED_EARNINGS.DTA includes data on years of education for a random sample of women aged 30 years in 2010.

(a) Use command mean education to give a 95 percent confidence interval for the population mean earnings of 30 year-old women.

(b) Now calculate the same 95 percent confidence interval using the confidence interval formula. You can get s from summarize education and ?.025,?1 using function invttail(n-1,0.025).

2. Population Mean and Variance

Let X denote annual health costs for an individual and suppose ? = 0 with probability 0.6, ? = 1,000 with probability 0.2 and ? = 9,000 with probability 0.2.

(a) Obtain ? = ?[?] from first principles.

(b) Obtain ? 2 = ?[(? ?) 2 ] from first principles.

(c) Hence find the standard deviation of X.

3. Mean and Variance of the Sample Mean

Let ? be the mean of a random sample of size 100 from a random variable that is distributed with mean 300, variance 1600, and a distribution that is not the normal distribution.

(a) Give the mean of ?.

(b) Give the variance and standard deviation of ?.

(c) Is ? likely to be normally distributed? Explain.

4. Coin Toss Example

Let ? = 1 if the toss of a six-sided die yields a five, and let ? = 0 otherwise. Then jQuery224016607416861192625_1587811867436[? = 1] = 1 6 and jQuery2240168348872003258_1587812115396[? = 0] = 5/6.

(a) Obtain ? = ?[?] from first principles.

(b) Obtain ? 2 = ?[(? ?) 2 ] from first principles.

(c) Give the Stata commands

clear

set obs 1000

set seed 12345

generate u = runiform()

hist u, start(0) width(0.1)

generate x = 0

replace x = 1 if u < 1/6

(d) Does u appear to be uniformly distributed on the interval (0,1)? Use a histogram.

(e) Compare the sample mean ?and sample variance ? 2 of variable ? in this sample to your theoretical answers in parts a-b.

(f) Obtain the histogram of variable x. Does x appear to be normally distributed?

5. Coin Toss Example - repeated samples

A computer experiment similar to the previous was run 400 times, yielding 400 samples that in the experiment were each of size 100. The sample mean ?for each sample is given in dataset AED DIETOSS.

(a) Obtain the descriptive statistics for the 400 values of ? . Are the mean and standard deviation what you expect? Explain.

(b) Obtain the histogram (or the kernel density graph). Is this what you expect? Explain.

6. Estimating production time A widely used technique for estimating the length of time it takes workers to produce a product is the time study. In a time study, the task to be studied is divided into measurable parts (subtasks), and each subtask is timed with a stopwatch or filmed for later analysis. For each worker, this process is repeated many times for each subtask to ensure a representative sample. Then the average and standard deviation of the time required to complete each subtask are computed for each worker. A worker's overall time to complete the task under study is then determined by adding his or her subtask averages. The data (in minutes) in TIMES.txt are the result of a time study of a production operation involving two subtasks, subtask 1 and subtask 2. Two works (A and B) were measured 7 times for each subtask.

a) Find the average time it took each worker to complete the overall task (subtask 1 plus subtask 2) under study.

b) For each worker, find the standard deviation of the time it took them to complete subtask 1.

c) In the context of this problem, what are the standard deviations you computed in part b are measurement of? In other words, what do they tell you about how the worker completes the task?

d) Repeat part b for subtask 2.

e) If you could choose workers similar to A or workers similar to B to perform subtasks 1 and 2, which type would you assign to each subtask?

7. Rental Market in Irvine

a) Download data for the Zillow Rent Index (ZRI) for Irvine, Orange County (California, not Florida), the state of California, and the United States from https://www.zillow.com/research/data/. (You don't need to show your work here.) Specifically, select data type "ZRI Time Series: Multifamily ($)".

b) Make a single Stata dataset that has ZRI time series for Irvine, Orange County, California and the United States in Stata.

c) Make a single time-series graph that proves the ZRI for Irvine, Orange County, California and the United States over time. Interpret the graph: describe the overall trends over time, say whether California looks different from the US, whether orange county looks different from California, and whether Irvine looks different from Orange County.

d) Calculate average ZRI in 2019 for Irvine, Orange County, California, and the United States.

e) Generate the time series for annual percentage change in ZRI for each geography. Plot them in a single time series graph, calculate summary statistics, and describe your findings. How fast has rent been rising in Irvine relative to Orange County, California, and the USA?