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PAST FINAL EXAMINATION 3

STAT7055 Introductory Statistics for Business and Finance

It is widely accepted that regularly flossing your teeth will decrease the chance of get- ting a cavity. Also, people who are generally more health conscious tend to floss more regularly. A study was conducted to investigate how often people flossed, how often people exercised and the relationship between flossing and exercising. A sample of 500 people were surveyed about their flossing and exercising habits and the resulting data is summarised in the following table (which lists the number of people falling into each category of flossing frequency and exercise frequency):

Flossing frequency (days/week)

		0	1	2	3	4	5	6	7
Exercise	Less than 1	23	32	22	21	17	15	14	10
frequency	Between 1 and 5	21	21	28	32	34	19	15	11
(hrs/week)	More than 5	12	11	16	24	33	31	21	17

(a) [4 marks] Test whether the proportion of people who floss 3 days a week is the same among people who exercise less than 1 hour a week and among people who exercise more than 5 hours a week. Clearly state your hypotheses and use a signif- icance level of α = 5%.

(b) [4 marks] Test whether the population proportions of people who exercise less than 1 hour a week, between 1 and 5 hours a week, and more than 5 hours a week are the same. Clearly state your hypotheses and use a significance level of α = 10%.

(c) [4 marks] For people who exercise more than 5 hours a week, test whether there are 3 times as many people who floss at least 4 days a week than people who don’t. Clearly state your hypotheses and use a significance level of α = 5%.

The more you floss, the less likely you are to see a dentist, which means that you will probably spend less on dentist fees each year. Ideally, we would like to go back and ask each of the 500 people how much they spent on dentist fees each year. Unfortunately, the study only had enough funding to go back and ask 2 people from each cell of the first table about their yearly dentist fees. The overall sample variance of the dentist fees of the 48 people was s² = 1592.212. In addition, a two-way ANOVA was performed on the data and the ANOVA table is displayed below:

Source	Sum of squares	Degrees of freedom	Mean squares	F
Flossing	27230.81	?	?	?
Exercise	4766.30	?	?	?
Interaction Error	17883.32 ?	? ?	? ?	?
Total	?	?

(d) [4 marks] Test whether there is an interaction between flossing frequency and exercise frequency. Clearly state your hypotheses and use a significance level of α = 5%.

(e) [3 marks] Test whether there is a difference in the mean yearly dentist fees between the different levels of flossing frequency. Clearly state your hypotheses and use a significance level of α = 5%.

(f) [3 marks] Test whether there is a difference in the mean yearly dentist fees between the different levels of exercise frequency. Clearly state your hypotheses and use a significance level of α = 5%.

Question 2 [13 marks]

Suppose I have 5 fair coins, two which have a head on both sides,  one which has a tail on both sides, and two which are normal (a head on one side and a tail on the other side).

(a) [3 marks] I shut my eyes, pick one of the 5 coins at random, and flip it. Find the probability that the lower face of the coin is a head.

(b) [3 marks] I open my eyes and see that the coin is showing heads (on the upper face). Find the probability that the lower face is a head.

(c) [4 marks] I shut my eyes again and flip the same coin again. Find the probability that the lower face is a head. Remember that I already saw a heads on the upper face on the first flip.

(d) [3 marks] I open my eyes and see that the coin is showing heads (on the upper face). Find the probability that the lower face is a head. Remember that I already saw a heads on the upper face on the first flip.

Question 3 [22 marks]

We have a coin which we suspect may not be fair. Suppose the coin is flipped 40 times and the number of times it comes up heads is 15.

(a) [4 marks] Test whether the probability that the coin comes up heads is not less than 0.5. Clearly state your hypothesis and use a significance level of α = 10%.

(b) [4 marks]  Calculate the probability of making a type II error for the test in  part

(a) at a significance level of α = 10%, if the true probability of coming up heads is 0.45.

(c) [3 marks] Suppose the true probability that the coin comes up heads was indeed

0.45. How many coin flips are required to estimate the true probability to within 0.1, with 99% certainty?

Suppose we have 10 of these unfair coins, where the probability that each coin comes up heads is 0.45. Just because we can, we decide to perform some more fun experiments with these coins. The first experiment is that we flip each of the 10 coins once. Each coin that comes up heads is flipped a second time. Let X be the number of heads that appears in the second round of flips.

(d) [4 marks] Calculate E(X) and V (X).

For the second experiment, we take two of these unfair coins and flip them. For the ith coin, define a new random variable Y_i in the following way: if the coin comes up heads, Y_i is equal to 1; if the coin comes up tails, we roll a fair six-sided die and Y_i is equal to the number that comes up. Let Z denote the sum of the numbers we get for each coin, that is, Z = Y₁ + Y₂.

(e) [4 marks] Calculate E(Z). (f) [3 marks] Find P (Z < 4).

Question 4 [7 marks]

Suppose X has a binomial distribution with parameters n and p. Recall that the variance

of the sample proportion, pˆ = ^X , is equal to V (pˆ) = ^p(1−p) .  If we were trying to estimate

n n

Question 5 [16 marks]

The effect of alcohol on the body appears to be much greater at higher altitudes.  To  test this theory, an investigator randomly sampled 12 people and randomly divided them into two groups of six. The first group was given 100cc of alcohol (in the form of free beer) at sea level. The second group was transported halfway up Mount Everest and given the same amount of alcohol. The blood alcohol level ( 100) of each person in each group was measured and the results summarised in the following table:

(a) [5 marks] Test whether the population variances of the alcohol levels in the two groups are equal. Clearly state your hypotheses and use a significance level of α = 5%.

(b) [5 marks] Test whether the population mean alcohol level from group 2 (15 000 feet) exceeds the mean from group 1 (sea level) by more than 1. Clearly state your hypotheses and use a significance level of α = 5%.

The investigator doesn’t fully trust the results from the test in part (a) and decides      he needs to collect more data. Suppose that he repeats the entire experiment 5 times. That is, he takes a new sample of 12 people each time, gives them free beer in the same manner described above and measures their blood alcohol levels. Each time he repeated the experiment, he calculated the sample variances of the blood alcohol levels in the two groups. The sample variances are listed in the table below:

(c) [6 marks] Based on this new data, test whether the population variances of the alcohol levels in the two groups are equal. You can assume that the population variances of these samples variances are the same in each group. Clearly state your hypotheses and use a significance level of α = 5%.

Question 6 [26 marks]

A study was conducted to determine what variables might be important in predicting the height of a professional basketball player. The Height (cm), Weight (kg), Wingspan (cm), Salary (million dollars) and Vertical Leap (inches) were recorded for 9 profes- sional basketball players. The data are summarised in the table below, along with some summary statistics.