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56 points
Statistics 108, Exam 2, Fall 2008
Name (print):_____________________
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not look at anybody else’s exam and I will take all necessary efforts to prevent others from seeing my
exam. The consequence of using additional test aids, copying from others, or allowing others to copy my
work will result in disciplinary action.
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Signature: _____________________
1. (3 pts) Suppose the probability of a random student being in debt is 0.25. Furthermore, the
probability of a student being in debt is 0.40 if the student is majoring in the liberal arts. That is,
P(debt)=0.25 and P(debt | liberal arts)=0.40. Are a student’s major and a student’s probability of
being in debt independent? Explain why or why not.
2. (4pts) Circle which one statement best describes the key part of the central limit theorem.
i.
As the sample size increases, the population becomes distributed more like the normal
distribution.
ii.
As the sample size increases, the population variance decreases.
iii.
As the sample size increases, the distribution of the sample becomes distributed like the
normal distribution.
iv.
As the sample size increases, the distribution of the sample means become distributed
more like the normal distribution.
3. (4pts) Circle which one statement best describes the key interpretation of a 95% confidence
interval for the mean.
i.
If 100 different random samples were taken from a population and the 95% confidence
interval calculated for each sample, we would expect the sample mean to be inside about
95 of the intervals.
ii.
A 95% confidence interval is an interval that contains 95% of the population values.
iii.
If 100 different random samples were taken from a population and the 95% confidence
interval calculated for each sample, we would expect the population mean to be inside
about 95 of the intervals.
iv.
A 95% confidence interval has a 95% chance of including the sample mean.
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4. Suppose the number of children in a family is distributed according to the following probability
distribution. The probability distribution for the number of children (k) is given below with the
exception of the probability for 2 children.
k
1
P(X=k)
0.3
cdf: P( X  k )
???
2
????
???
3
0.2
???
4
0.05
???
a) (3pts) Fill in the table for P(X=2).
b) (3pts) Fill in the table for the cumulative distribution function (cdf) column.
c) (3pts) Calculate E(X); i.e., the mean number of children in a family.
5. Suppose you toss 7 fair coins, where the outcome of one toss is independent of another.
a) (4pts) Calculate the probability of getting exactly 4 heads and 3 tails. Show your work.
b) (2pts) What is the expected number of heads. Show your work.
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6. Suppose X is a random variable from a uniform distribution where the minimum value of X is 0
and the maximum value of X is 5.
a) (3pts) Draw the probability density curve for X and label the x and y axes, in particular
the units on the axes so that the height and width of the curve is easily distinguishable.
b)
(2pts) Calculate P(X < 3.5).
7. Suppose Z is distributed according to the standard normal distribution.
a) (4pts) Calculate P(-1 ≤ Z ≤ 1.50). Show your work.
b) (3pts) Use the table to find the 33rd percentile of Z.
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8. Suppose X is distributed according to the normal distribution with a mean of 50 and a standard
deviation of 8.
a) (3pts) How many standard deviations is 46 from 50? Below or above? Show your work.
b) (2pts) Calculate P(X ≤ 46). Show your work.
c) (2pts) Calculate P(X > 46).
d) (4pts) Suppose 16 values of X were randomly sampled and the mean, 𝑋̅, calculated. Find
P(𝑋̅ ≤ 46). Show your work.
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9. Suppose the mean weight of 16 adult male golden retrievers (a breed of dog) resulted in a sample
average of 32 kg. Assume the population standard deviation is 5 kg.
a) (4pts) Calculate a 95% confidence interval for the mean weight of adult male golden
retrievers. Show your work.
b) (3pts) Suppose you wanted to estimate the mean with a margin of error no larger than
±1kg. What is the minimum sample size needed? Show your work.