Assignment: Probability and Sampling Distributions

In this assignment, you will be answering questions about probability and sampling distributions by hand and using Microsoft Excel. 

1) A normal deck of playing cards has 52 cards, divided into 2 colors (black and red), and 4 suits: Hearts (red), Diamonds (red), Clubs (black), Spades (black).  Each suit has the number cards 2 through 10, then the "face" cards Jack, Queen, King, and Ace.  So for example, there are 4 aces:  Ace of Hearts, Ace of Spades (the ACE of SPAAAAADES!), Ace of Diamonds, and Ace of Clubs.

A) What is the probability (written as a percent) of drawing a Diamond card?

B) What is the probability (written as a proportion) of drawing a face card?

C) What is the probability (written as a percent) of drawing either a diamond card OR a heart face card?

D) You draw one card, put it back in the deck (sampling with replacement), and draw a second card.  What is the probability (written as a proportion) of drawing two Aces in a row?

E) You draw two cards without putting them back in the deck (sampling without replacement).  What is the probability (written as a percent) of drawing two heart cards?

2) You are playing craps, which is a dice game where you roll two six-sided dice in a single throw.

A) Write out all possible combinations of the two dice (i.e., your sampling distribution).  For example, if you roll a 1 on the first die and a 1 on the second die, your sample is a 1 - 1.

B) What is the probability (written as a percent) of getting a total of 7 on a single throw?

C) What is the probability (written as a proportion) of getting snake-eyes (two 1's) on a single throw?

D) What is the probability (written as a percent) of NOT getting a 7 or an 11 on a single throw?

3) You are playing Dungeons and Dragons.  One of the most common gaming dice is a 20-sided die.

A) A "critical hit" occurs on a roll of 20.  What is the probability (written as a percent) of getting a critical hit on a single roll?

B.) You are fighting 3 goblins.  You can attack each one of them once, by rolling the 20-sided die once per goblin.  You need a 13 or higher to hit a goblin.  What is the probability (written as a proportion) of hitting all three goblins (i.e., Goblin 1 AND Goblin 2 AND Goblin 3)?

4) In the normal distribution, what is the probability (written as a proportion) of pulling a sample that is greater than z = +1.96 OR less than z = -1.96?

5) Use your data from the M&M portion of Lab 5 to answer the following questions.

A) For your individual sample, what is the probability of drawing a blue M&M?

B) For your individual sample, what is the probability of drawing a yellow M&M?

C) For your pooled sample, what is the probability of drawing a blue M&M?

D) For your pooled sample, what is the probability of drawing a yellow M&M?

E) For the class sample, what is the probability of drawing a blue M&M?

F) For the class sample, what is the probability of drawing a yellow M&M?

G) For the population data (the data provided by Mars company), what is the probability of drawing a yellow M&M?

H) For population data (the data provided by Mars company), what is the probability of drawing a yellow M&M?

I) Imagine that Prof. Murphy randomly selects 5 blue M&M's from the Class Sample and eats them (sampling WITHOUT replacement), how does this affect his probabilities for the next sample he takes?  (i.e., think about what it does to the probability of getting a blue M&M, vs. a different type of M&M.)  Give specific calculations.

J) Now, let's say that you select 5 blue M&M's from your Pooled Sample and eat it.How does this affect your probabilities for the next sample you take?  (i.e., compare this to answer 5a.  Does this error affect the odds MORE or LESS?)  Give specific calculations.

K) Finally, let's say that you select 5 blue M&M's from your small sample and eat it.How does this affect your probabilities for the next sample you take?  (i.e., compare to 5a and 5b.  Does this error affect the odds MORE or LESS?)  Give specific calculations.

6) Assume that the mean weekly study time for the entire student body of a large university is 25 hours, with a standard deviation of 15 hours.

A) What is the mean of the sampling distribution of the mean (n = 100)?

B) What is the standard error of the sampling distribution of the mean (n = 100)?

7) A random sample is obtained from a population with μ = 100 and σ = 20.

A) How much error would you expect on average (i.e., Standard Error of the Mean) between the sample mean and the population mean for a sample of n = 4 scores?

B) How much error would you expect on average between a sample of n = 16 scores?

C) How much error would you expect on average for a sample of n = 100 scores?

Microsoft Excel

8) In this problem, we will be finding the sampling distribution for a given population and sample size, and calculating the standard error.  Imagine a very simple population consists of the following 5 observations:  2, 4, 6, 8, 10.  We will be taking a sample of n = 2.

a) Open Excel.

b) In cell B1, type in Population, and make it bold.

c) Type in the values for our population in column B.

d) In cells A8 through A10, type in Pop'n Mean, Pop'n SD, and SEM.

e) In cells B8 and B9, find the mean and population standard deviation for our population.  In cell B10, divide the population SD (cell B9) by the square root of our sample size (n = 2).  Note that this is the formula for the Standard Error of the Mean.

f) In columns D and E, starting in cells D2 and E2, list all possible samples for n = 2, using sampling with replacement.  For example, the first possible sample would be 2 (in cell D2) and 2 (in cell E2).  There will be 25 possible samples.

g) In cell G1, type in Sample Mean and make it bold.

h) In cell G2, find the mean of the sample (i.e., the mean of cells D2 and E2).  Drag this formula down for all of our possible samples.

i) In cells F28 and F29, type in Mean of Means and SEM.  Make these cells bold.

j) In cell G28, find the mean of each sample.  This value represents the mean of our sampling distribution, and should also be equivalent to the population mean.

k) In cell G29, find the population standard deviation of the sample means.  This value represents the standard error of the mean, or how spread out the sample means are.  This should match with the standard error of the mean you calculated in cell B10.

l) In cells J1 and K1, type in Possible Sample Means and f, and make these cells bold.

m) In cells J2 through J10, list the possible sample means.  Don't repeat a number.  There should be 9 possible sample means.

n) In cells K2 through K10, find the frequency of each possible sample mean.  Notice how there are some samples that are very common, and some samples that are very rare; the sampling distribution is normally distributed

o) Save your Excel file as Lastname_Lab6_SamplingDistribution. Upload your Excel file to Canvas.

Attachment:- Data.rar