Statistics Homework

Name: __________________________________________________

JHU—Advanced Academic Programs

Graduate Program for Applied Economics

Statistics 440.605.82

Summer 2017

HW—UNIT #4 DUE TUES June 6th by 11:59PM (EST)

Please read each question carefully and answer it completely. “Extra work” not explicitly needed can be copied into the end of document in the Appendix. If an answer is incorrect, including work in the Appendix might allow for minimal “partial credit.” (Note that although you are not always required to “show work” on the homework assignments, you may be required to do so on exams.)

Unless otherwise instructed, please round your answers to two decimal places. For these problems you can assume an infinite population.

#1. Central Limit Theorem. In an EXCEL file to accompany this assignment, you are given a worksheet with 25 samples drawn (n = 20) from a continuous uniform distribution for x defined between 0 and 20.

(a) What are µ and σ for this uniform distribution, U[0,20]?

(b) What are the pdf, f(x), and cdf, F(x), for this uniform distribution?

(c) Find the sample means for those 25 samples and fill in those values for “X-bar” () in the table found in worksheet “table” in the EXCEL file. Note that you now have one column of 25 observations for a sample from a U[0,20] marked “X” and 25 sample means computed from samples drawn from that distribution. Create a “Descriptive Statistics” table of these two variables. Paste your “Descriptive Statistics” table below:

(d) Highlight three differences between these two variables in your table, i.e. Descriptive Statistics values for “X” versus “X-bar.”

(e) Based on theory, what should be the mean and standard deviation of taken from this distribution with this sample size, n?

(f) What about your sample results for suggest that the distribution is more “normal” than the data for X?

(g) Fill in the following table to help think about what the samples suggest about the underlying shape of these distributions. Comment on what this demonstrates.

	Data for X	Data for “X-bar”
% of the observations between 4 and 6
% of the observations between 2 and 4
% of the observations between 6 and 8

(h) What does the Central Limit Theorem suggest would make the distribution for “even more normal?” (Hint: consider p. 315)

#2. Textbook p. 322, #29. [Questions begins “The mean preparation fee for H & R Block …”] Please complete (a) – (c) using a table similar to last unit’s assignment. Note that I have added a column for you to describe the mean and std dev of the distribution for . You can use F(X) for the cdf.

	Please fill in the mean and std dev for	cdf notation	EXCEL commands	Answer (for probabilities, three decimal places)
(a)	mean = std dev =
(b)	mean = std dev =
(c)	mean = std dev =

(d) (from textbook)

#3. Textbook p. 337, #51. Question begins: “A market research firm conducts telephone surveys …” Please solve this probability using two methods: (a) using the distribution for and (b) “reverse engineering” this into a binomial distribution problem so you can solve for the “exact” probability using the =binom.dist() command in EXCEL. [Note that defining the problem through proportions is another technique to use a normal approximation for yes/no (qualitative) binomial-type of experiments.]

Comment: For those of you who have been able to install “Data Analysis,” note that you can use the ADD-IN to generate random numbers. Below is a screen shot of how I used the command window to generate the samples for this assignment (note that the command requires a “seed,” a four digit number I filled in off the top of my head, which starts the generation process):

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