Create a frequency distribution and relative frequency distribution of the observations.

INSTRUCTIONS TO CANDIDATES

ANSWER ALL QUESTIONS

Question #1: Every morning James randomly chooses one shirt out of his wardrobe. He only owns shirts that are either red or green. Cecelia wants to find the total number of shirts and how many of each colour he owns. Instead of asking him she keeps a daily record of the color of James’s shirts. (Assume that James does laundry every day, so the number of types of shirts remain the same.)

A. Create a frequency distribution and relative frequency distribution of the observations.

B. Give an estimate for the probability that James will wearing a red shirt that day, call this result 𝑝̂.𝑝̂ =

C. One day Cecelia asked James for the number and color of shirts he owns. He claims that out of the 35 shirts he owns 14 of them are green. Should Cecelia believe his claim? Justify your answer.

D. For the next 14 days James will keep to his pattern of choosing a shirt. Let 𝑋 be a random variable defined as:

𝑋: {# 𝑜𝑓 𝑑𝑎𝑦𝑠 𝑜𝑢𝑡 𝑜𝑓 𝑛𝑒𝑥𝑡 14 𝑡ℎ𝑎𝑡 𝐽𝑎𝑚𝑒𝑠 𝑤𝑒𝑎𝑟𝑠 𝑔𝑟𝑒𝑒𝑛 𝑠ℎ𝑖𝑟𝑡. }

What distribution does X follow? Justify your answer by showing it follows ALL the criteria for your choice of distribution.

E. Which is more likely:

a. James wears a green shirt 5 out of the next 14 days given Cecelia’s estimate is true.

b. James wears a green shirt 5 out of the next 14 days given his claim is true.

F. Let 𝑌 be defined as

𝑌: {# 𝑜𝑓 𝑡𝑖𝑚𝑒𝑠 𝐽𝑎𝑚𝑒𝑠 𝑤𝑜𝑟𝑒 𝑎 𝑟𝑒𝑑 𝑠ℎ𝑖𝑟𝑡 𝑤ℎ𝑖𝑙𝑒 𝑏𝑒𝑖𝑛𝑔 𝑜𝑏𝑠𝑒𝑟𝑣𝑒𝑑 𝑏𝑦 𝐶𝑒𝑐𝑙𝑒𝑙𝑖𝑎}

why is 𝑃(𝑌 = 90) = 0 ? 

Question 2: Suppose you have the following population: {24, 36, 48, 60, 72, 84, 96}

A.What is this population's 𝜇 = 𝜎 =  𝑁 = 

B.Suppose you selected a sample of size n=2 from this population. What is the number of possible combinations of sample size n= 2?

C.In the table below list all the possible combinations for samples of size n=2, and their sample means, 𝑥̅ :

Table 1

Sample Size n=2	Sample mean, 𝑿̅
{24, 36}

D. What is the probability of selecting the combination of 24 and 36? 

In fact, each of the sample combinations has what probability of being selected? 

E. What is the population mean and population standard deviation of the sample means in Table 1?

𝜇𝑥̅  =𝜎𝑥̅ =  

F. Using the Empirical Rule what interval, centred by the mean, will contain 68% of the observations of the sample mean? What about 95% of all observations?

G. Generate a table that displays the probability distribution for the values of the sample mean (Hint: a relative frequency distribution can function the same way)

H. Generate a probability histogram for the sample mean

Attachments:

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