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Mathematics 109 final test
Tuesday, December 13, 2011
please show any relevant work to get credit for each problem
1. To determine President Obama’s re-election chances, I decide to telephone all my friends
and ask whether they plan to vote for him. What kind of sampling is this?
2. What method of data collection would best be used to determine whether large doses of
vitamin C help to prevent catching a cold ?
3. Identify each quantity as a parameter or a statistic:
• p̂
• x̄
• s
• µ
4. What is a type I error ?
5. How many distinct “words” (they don’t have to make sense) can be formed by re-arranging
the letters in OBAMA ?
6. Calculate how many different committees of 12 people can be formed from a group of 20
people.
7. Calculate the z-score for x = 13 cm when the mean µ = 19 cm and the standard deviation
σ = 3 cm.
8. A sample of four individual heights: 150, 160, 156, and 170 (in cm) is selected from a
larger group.
Calculate from this small data sample
(a) the mean
(b) the median
(c) the range
(d) the standard deviation
page two
9. Suppose that in a certain group of 100 people, there are 24 tall men and 22 short men,
and 22 tall women and 32 short women. If I pick one person at random from this group,
compute the probability that
(a) this person is female
(b) this person is male or short
(c) this person is tall, assuming that they are female
(d) If a pair of people is randomly chosen from the group of 100, what is the probability
that both are short ?
10. Suppose the heights of trees in a large forest are approximately normally distributed,
with a mean height of µ = 20 meters and a standard deviation σ = 6 meters.
(a) What percentage of the trees are 26 meters or higher ?
(b) Calculate the third quartile for tree heights.
11. Suppose that after a risky medical procedure, each patient has a 60% change of survival.
(a) If five patients have this procedure, what is the probability that none of them survive?
(b) If seven patients have this procedure, what is the probability that exactly four of them
survive?
(c) if nine patients have the procedure, what is the probability that at most eight survive?
12. Suppose that IQ scores of students in a large school district are normally distributed,
with a mean of µ = 105 and a standard deviation σ = 15.
Suppose that all possible samples of 100 students are drawn randomly out of the school
district, and the mean IQ is calculated for each sample
• what is the mean of all the sample means?
• what is the standard deviation of the sample means?
• what is the probability that a sample mean IQ is greater than 107 ?
13. Suppose that 60% of the time, it takes a commuter 15 minutes to get to work, 30% of
the time it takes 25 minutes, and 10% of the time it takes 50 minutes. Calculate the mean
(or weighted average) of these commute times.
page three
14. Annual income in a randomly selected group of 1600 people is studied: the mean annual
income was found to be x̄ = $60, 000, with standard deviation s = $8, 000. Construct a 90%
confidence interval for the mean annual income.
15. A claim is made that the mean income of people in a first city (A) is greater than the
mean income of people in a second city (B).
• state the null hypothesis and the alternate hypothesis
Samples of 2500 randomly selected people are chosen from each city. The income statistics
found were:
city A : x̄1 = $70, 000 and s1 = $5, 000
city B: x̄2 = $69, 400 and s2 = $4, 000
At level of signifigance α = .01, can we support the claim that the mean income of people
in the first city is greater than the mean income of people in the second city?
16. Construct a histogram for the following data:
( break it into three categories – between 30 and 35, between 35 and 40, and between 40
and 45)
31, 31, 32, 32 , 33, 33, 36, 36, 39, 41, 42, 43, 44, 44, 44, 44
17. Use your histogram to construct a cumulative frequency graph (ogive) for the data of
the previous problem
(use the same three categories as for the histogram)
page four
18. Given the data below
12
10
8
6
4
2
0
0
2
4
6
8
10
12
14
(a) draw a best-fit line ( no need to calculate the least-squares line)
(b) find an equation for your best-fit line
(c) estimate the value of the (Pearson) correlation coefficient for this set of data.