Question 1: An urn contains 50 beads. The beads are identical in shape and have one of two colors: blue and orange. We would like to estimate the proportion p of blue beads. We select without replacement a sample of 10 beads. The relevant statistics is the sample proportion p^ of blue beads (i.e., the number of blue beads in the sample divided by 10.
For the purpose of the simulation exercise, we will assume that the box contains exactly 15 blue beads or, equivalently, the proportion of blue beads is p = 0.30..
i) Select 100 samples of size 10 from the box.
ii) Compute the sample proportion p^ of blue beads for each of the 100 samples found in (i).
iii) Make a histogram of the values of p^ found in (ii) (that is the approximate sampling distribution of p^.)
iv) Find the summary statistics of the 100 values of p^.
v) Base yourself on the histogram and the summary statistics to describe the approximate sampling distribution of p^.
vi) Is p^ an unbiased estimator of p? Hint: Evaluate the difference between p = 0.30 and the mean value.
vii) Based on the histogram, estimate the probability that p^ > 0.40 and the probability that p^ < 0.20?
Question 2: Repeat question 1, but this time we select without replacement a sample of size 20.
Question 3: Assume your boss has asked you to estimate the proportion of blue beads in the urn described above. Based on your findings in questions 1 and 2, which of the two sampling distributions would you prefer to work with. Explain your choice.
In questions 4-6, we have R simulate confidence intervals for a normal population mean.
Question 4: (R)
i) Generate 25 samples of size 16 from a normal population with mean and standard deviation σ = 100. (nothing to take to your Word file)
ii) For each sample found in a), construct a 95% confidence interval for the population mean.
iii) Verify by hand and for sample 1 only the results obtained by R. Note that the sample mean is the midpoint of the confidence interval.
iv) How many intervals contain μ. Would you expect all 25 confidence intervals to contain μ? Explain your answer.
Question 5: (R) - Repeat question 4, but this time use an 80% confidence level.
Question 6: (R) - Based on the simulations you conducted in questions 4 and 5, what are the differences between 80% and a 95% confidence intervals for a population mean?