Questions -

Q1. The trade volume of a stock is the number of shares traded on a given day. Two separate random samples, each for 40 trading days in 2015, are obtained and the volume of Harley-Davidson stock traded (in millions of shares) for each day is recorded. The results are stored in the file Harley.

(a) Use the data to compute two different point estimates for the mean number of shares in Harley-Davidson stuck traded per day in 2015.

(b) Using the first sample, find a 90% confidence interval for the mean number of shares in Harley-Davidson stock traded per day in 2015.

(c) Using the second sample, find another 90% confidence interval for the mean number of shares in Harley-Davidson stock traded per day in 2015.

(d) Interpret the confidence intervals you found in parts (b) and (c). Explain why they are different, and what they tell you about the mean number of shares in Harley-Davidson stock traded per day in 2015.

Q2. The performance of a stock is measured by its one-year rate of return, which is the ratio of money gained (or lost) to the amount of money invested; Brokerage firms may use this measure of performance to make recommendations to a potential investor. The performance of 15 stocks (as measured by their rate of return over 2015) recommended by similar firms in the month of February 2016 have been recorded. The results are stored in the file Returns.

(a) Using the data given, find a 96% confidence interval for the mean rate of return for the stocks recommended in February of 2016.

(b) Using the same data, find a 98% confidence interval for the mean rate of return for the stocks recommended in February of 2016.

(c) How does the interval you found in part (b) compare to the interval found in part (a)? Give an explanation for why the two intervals are related in this way.

(d) Investors commonly use the standard deviation of return rates as a measure of risk for investing, i.e., a larger standard deviation implies greater risk to the investor_ Using the data find confidence interval for the standard deviation of the rates of retort for the stocks recommended in February of 2016.

(e) In finding the above confidence intervals, what, if any, assumptions are you making? Are these assumptions supported by the given data?

Q3. Human eye color is classified using live catescrries1 brown, blue, grey, green, and hazel. According to genetics, brown should be the most common eye color, and green the least common. A random sample of 60 adults (30 females and 30 males) is obtained and the eye colors of each individual are recorded (as well as the sex of each). The results are stored in the tile Eyecolor.

(a) Perform Exploratory Data Analysis on the data, i.e., construct appropriate visual summaries and compute appropriate numerical summaries.

(b) Treating the sample of 30 females and the sample of 30 males as two separate random samples, find a 95% confidence interval for each the proportion of females with brown eyes as well as the proportion of males with brown eyes. Based on these intervals, do you think there is a difference between the two proportions based on sex? If so, which sex has a higher proportion of brown eyes?

Q4. A particular Wal-Mart stare wants to analyze the amount of time customers must wait in line at the customer service desk. In particular, they want a model for the population of wait titles and an estimate of the average wait time. A random sample of n = 478 wait times (in minutes) over the past month has been obtained- The results are stored in the tile Service.

(a) A consultant hired by the store recommends using an exponential distribution to model the population of wait times. Based on the provided sample, do you agree? In particular, do you agree that wait times are not normally distributed?

(b) Assuming the population of wait times follows an exponential (λ) distribution; find the maximum likelihood estimate of λ given by the provided sample.

(c) Using the MLE of λ, give a point estimate of the average wait time.

(d) Now use the Bayesian approach to estimate the parameter λ, where the prior distribution of λ is given by a gamma (α, β) distribution.

i. Suppose that similar studies at other Wal-Mart stores produced wait times with an average of 5 minutes and a standard deviation of 3 minutes. Given this information, select values of the parameters for the prior distribution of λ. Note that if random variable X ∼ gamma(α, β), then E[X] = α/β and Var(X) = α/β².

ii. Give the posterior distribution of λ based on the provided sample.

iii. Give a Bayesian point estimate of λ.

iv. Give a 95% credible interval for λ.

v. Using the Bayesian estimate of λ you found in part iii, give a point estimate of the average wait time.

(e) Compare the two point estimates of λ you found. Discuss the advantages/disadvantages of each approach. Which do you prefer?

Needs to be completed using RStudio. All required files attached herewith. change attached .xlxs file to .csv

Attachment:- Assignment.rar