STATISTICS FINAL EXAM

Write neatly, big enough to see, show the formulae you use:

1) If we draw 3 cards from a well-shuffled, fair deck of 52 playing cards,
what is the probability of getting:

 a) 3 queens b) 2 hearts and 1 diamond c) a 6 and 2 kings

(6)

2) The probability a company will make a profit or break even is 5/9.
What are the odds that the company suffers a loss?

(4)

3) Two options a car buyer may select are power windows (W) and power brakes (B).
A dealer's records show that the probability that a buyer selects power windows is 0.50
and the probability he selects power brakes is 0.60. The probability a buyer selects
power windows given that he's already selected power brakes is 0.70.
Find: (Show the formula you use.)

a) The probability that a buyer bought both options.
b) The probability that a buyer who bought power windows also bought
c) The probability that a buyer bought either option.
d) The probability that a buyer didn't select either option.
e) Use the right formula to determine if events (W) and (B) are independent.

(10)

4)

a) State the conditions that must be satisfied in order for a function to serve
as a probability distribution of a random variable.

(3)

b) Check whether f(x) = 4Cx /16 for x = 0, 1, 2, 3, and 4, can serve as the
probability function of a random variable. If so, list the values of the probability
distribution of this random variable, find the mean and standard deviation.

(8)

5) We intend to use the mean of a random sample of size n = 64 to estimate the mean
of an infinite population with
r = 20.
What is the probability that the error will be ± 5 using:

 a) Chebyshev's theorem b) the Central Limit Theorem ?
(6)

6) Records show that 25% of the cars manufactured at a certain plant need repairs
in the first 3 years of operation. IBM buys 90 of these cars.
Use the Normal Approximation of the Binomial Distribution to find the probability
that in the first 3 years of operation:(make a diagram, state µ and
r.)

a) less than 24 of the cars need repairs.
b) between 20 and 23 (inclusive) of the cars need repairs.

(6)

7) Random samples of size 2 are selected from the finite population consisting
of the numbers 3, 5, 7, 9, 11, 13.

a) Find the mean and standard deviation of this population.

(2)

b) List the 15 possible random samples (n = 2) that can be selected from
this population and calculate their means.

(5)

c) Use the results of part (b) to construct the sampling distribution
of the mean of these samples.

(2)

d) Calculate the mean and variance of the probability distribution in part (c)
and compare them with the results obtained in part (a).

(4)

8) A company tests a group of applicants for a job in data entry.
The times taken by the applicants to type a passage were normally distributed
with µ = 90 seconds and s = 12 seconds. If the company considers only the fastest 10%
of these applicants, what is the slowest time scored by an applicant that the company
will consider for the job? (diagram!)

(5)

9) The police department of Baie Comeau reports that 375 of 500 randomly
selected burglar alarms received at the station are false alarms.

a) What is the standard deviation of an estimate based on this data?

(2)

b) Construct a 90% confidence interval for the population proportion of false alarms.

(5)

10) In a study of the income of farming households in Ontario, the standard deviation is estimated to be \$3,000. How large a sample is required to estimate the mean income
within \$200 with a 95% level of confidence?

(4)

11) An anthropologist is studying the heights of the adult members of an ancient population. Conventional theory holds that the mean height of adult men from this population
is 56 inches. A sample of 12 men showed a mean of 53.5 inches with a
standard deviation of 2 inches.
Can the anthropologist conclude that the mean height is less than 56 inches?
Perform the test at the 0.05 level of significance.

(6)

12) A medical researcher claims that there is a significant difference between
the mean lung capacity of smokers and non-smokers.
In a test, 30 non-smokers have a mean lung capacity of 5 litres with
a sample standard deviation of 0.3 litres, and 40 smokers have a mean lung
capacity of 4.5 litres with a sample standard deviation of 0.4 litres.
If
a = 0.01, is there sufficient evidence to conclude that lung capacity
is greater for non-smokers?

(6)

13) 60 Boy Scouts were treated to ice cream at a pathetic little ice cream parlor
that had only 3 flavors: chocolate, vanilla and strawberry. If 24 chose chocolate,
17 chose vanilla, and the rest took strawberry, test at the 1% level of significance
if there is a flavor preference amongst these Boy Scouts.

(5)

14) A manufacturer has test-marketed a new CD player. Random samples of students
at a university were asked to try the product for 4 weeks and then were asked
if they would buy the product. The results are as follows:

 science arts medicine TOTAL WILL BUY 13 12 25 50 WON'T BUY 87 88 275 450 TOTAL 100 100 300 500

Would you conclude that the respondents' preferences are independent of university program? Conduct the test at a = .05

(6)

15) A recent report from the company that makes M&M chocolate covered peanuts
claims that every bag contains 30% brown, 20% green, 20% red, 20% yellow
and 10% orange candies. You buy a 1 kg. bag which contains a total of 188 candies
of which 67 are brown, 24 are green, 51 are red, 22 are yellow and 24 are orange.
Test at the 1% level of significance if your bag has the same color proportion
as claimed in the report.

(5)

TOTAL (100)

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