Q.1 The following is a stem–and–leaf plot shows of the cost of grocery purchases (to the nearest dollars) for 50 shoppers.

Q.2 Are better quality detergents more costly?  Penny-pinchers often assert that brand-name cleaning products are no better than their “no-name brand” alternatives; that is, the consumer is simplying paying more for the name brand’s packaging and marketing.  Researchers collected data from housewives and househusbands who regularly perform family laundry duties.  The researchers asked these customers to name and rate the quality of the product they currently use, and then recorded the selling price of the products on Amazon. Here is a plot and summary of the data:

25       30       35      40      45       50       55       60

Rating

Let the sample correlation of coefficient of correlation of Price and Rating is 0.6708.

(a) Using the graphical and numerical summaries, summarize the distribution of Price and Rating indi-

vidually, and describe the relationship between them.

(b) Suppose that we estimate the relationship between Rating and Price using a simple linear regression.

Estimate the coefficients of the straight line, and write down the equation for this fitted line.

(c) Below is a Normal qq-plot. Do you have any concerns about the normality?

Normal Q−Q Plot

−2              −1               0                1                2

Theoretical Quantiles

(d) What is your predicted Price when Rating is 55.

(e) Find and interpret R2 .

Q.3 An fisherman has been recording his success at fishing at his local lake in different weather conditions for many years. In particular, he has been studying what type of bait is more successful for catching fish

and has produced the following table.

Live Worm Bait

Sunny Weather

Cloudy Weather

Rainy Weather

(a) What is the probability the fisherman’s next fishing trip will be on a sunny day? (b) Is the type of bait independent of weather?

(c) If the fisherman goes on fishing on a rainy day, what is probability of catching a fish with a seaweed bait?

Q.4 Suppose A and B are two events in sample space S satisfying P(A) = 0.5 and P(B) = 0.6.

(a) If P(A|B) = 0.4, what is P(B|A)?

(b) If P(AB) = 0.1, what are P(A|B) and P(B|A)?

Q.5 It is estimated that 50% of emails are spam emails.  Some software has been applied to filter these spam emails before they reach your inbox.  A certain brand of software claims that it can detect 99% of spam emails, and the probability for a false positive (a non-spam email detected as spam) is 5%. Now if an email is detected as spam, then what is the probability that it is in fact a non-spam email?

Q.6 A black box contains 4 balls numbered 1–4.  Randomly pick 2 balls without replacement from the black box.  Let X be the larger number printed on the two balls.  For example, if we pick balls with numbers 1 and 3, then X = max{1, 3} = 3.

(a) Fully describe the distribution of X – present the distribution by listing all the possible values and the

corresponding probabilities of X .

(b) Based on your answer to Question (a), calculate E(X) and Var(X)

(c) Define two events

A = {X is an even number}

B = {X ≤ 3}

Are A and B independent? Justify your answer by calculation.

Q.7 Random variable Y follows a normal distribution with population mean 3 and population standard deviation 1.5. Find the value of an unknown constant t such that P(−t ≤ Y ≤ t) = 0.80

(Keep 3 decimal digits for your final answer)