Name:
Math 17 – Introduction to Statistics
PRACTICE First Midterm Exam
Instructions:
1. Show all work. You may receive partial credit for partially completed problems.
2. You may use calculators and a one-sided sheet of reference notes, as well as the provided table.
You may not use any other references or any texts.
3. You may not discuss the exam with anyone but me.
4. Suggestion: Read all questions before beginning and complete the ones you know best first.
Point values per problem are displayed below if that helps you allocate your time among
problems.
5. Good luck!
Problem
1
2
3
4
Total
12
11
12
15
50
Points Earned
Possible Points
1. Fifty-five parents of grade school children were interviewed regarding the breakfast habits of their
children. Two questions of interest were “Do your children eat breakfast?” and “What was your child’s
average grade on the state standardized test?” Responses to both questions were recorded as yes or no
for breakfast and values between 0-10 for grade. Grades for the breakfast and no breakfast groups
were summarized using a boxplot.
a. Is this an experiment or an observational study?
b. The response variable in this situation is
_______________ while the explanatory variable is
________________.
c. Breakfast is an example of a (choose one)
categorical
quantitative
variable.
d. Based on the boxplots, which breakfast group
performed better on the standardized test? How
can you tell?
e. What is (approximately) the IQR for the grade scores of children who do eat breakfast?
f. No outliers are visible on the boxplot. Explain why there are no outliers plotted specifically for the
group of children who do not eat breakfast.
g. The lowest score for the group of children who do eat breakfast is roughly equal to the
________________ for the group of children who do not eat breakfast.
h. If some of the selected parents for the original sample declined to answer both questions, what sort
of bias would have occurred?
selection bias
response bias
nonresponse bias
2. Julia and Dennis have collected a data set from elementary Grade\Pref Vanilla
school children at a summer camp on their ice cream
First
12
preference. They recorded each student’s grade
Third
48
(first,third,fifth, no second or fourth grade students) and
Fifth
24
their ice cream preference (choosing between vanilla and
Total
chocolate). The following table summarizes the results. You
may treat the probabilities resulting from the table as population probabilities.
Chocolate
4
16
8
Total
a. Compute the following probabilities:
i. The probability a randomly selected summer camp child is in the third grade
ii. The probability a randomly selected summer camp child preferred chocolate ice cream
iii. The probability that a randomly selected summer camp child is in the fifth grade given that he/she
prefers chocolate ice cream
iv. The probability that a randomly selected summer camp child is in the third grade given that he/she
prefers vanilla ice cream
b. Are third grade status and ice cream preference independent? How can you tell?
c. If you choose 2 summer camp children at random (assume random sample), what is the probability
that at least one is in the third grade?
d. Assume the summer camp children have to wait in line for components for Smores at the campfire.
Assume the distribution of waiting times is uniform from 3 minutes to 15 minutes. Let X be waiting time.
i. What is the expected value for waiting time in line?
ii. What is the probability a child waits between 3 and 6 minutes for their Smore components?
iii. Name another continuous distribution besides a uniform distribution.
3. A researcher studying the heights of certain shrubs in the forest
has collected a sample of 50 shrub heights in feet (continuous
numerical variable). However, he is unsure of how to proceed with
any analysis, but he makes a histogram and a Q-Q plot of the data.
a. What would you tell the researcher to do as a preliminary
analysis? (Be specific – what graphs, etc.)
b. The sample standard deviation of shrub heights was 1.34 with a
sample mean of 4.32. Interpret the standard deviation.
c. Based on the two plots the researcher made, what can you
conclude about the shape of the distribution of the shrub heights?
Why?
d. The researcher wants to examine the sample mean shrub height. What is the distribution of the
sample mean shrub height for a sample of size 50 if the population mean shrub height is 4 and the
population variance is 2? What result allows you to provide this distribution?
e. It turns out that the researcher’s original sample of heights was based on shrubs in a 1 km square
block within a biological research zone. The research zone has 16 such square kilometer blocks with 4
blocks at each of four altitudes. The researcher plans to obtain a larger sample of shrub heights, say 256
heights. From previous research, you know there is an effect of altitude on shrub height. Describe a
sampling plan that you might use if you were the researcher and you wanted to deal with this effect
when sampling.
4. Vitamin D deficiency has recently re-entered the public health scene as a serious problem among
American adults. The current recommendation is to have 400 IU/day of Vitamin D. Assume that in a
local community, the amount of Vitamin D ingested has a normal distribution with population mean 400
and standard deviation 40.
a. Ted and Janice are a happily married couple living in this community. Ted ingests 350 units of Vitamin
D daily while Janice ingests 420 units a day. Who has the more unusual Vitamin D intake? How can you
tell?
b. 20% of members of this community ingest under what amount of Vitamin D daily?
c. Assume that nationwide, 80% of adult Americans are Vitamin D deficient. A random sample of 60
adult Americans is selected.
i. What is the sampling distribution of the sample proportion of adult Americans who are Vitamin D
deficient for a sample of size 60? Include the check that this distribution is valid. (Give distribution type,
etc.)
ii. What is the probability that a sample of size 60 gives a sample proportion less than .75 as the
proportion who are Vitamin D deficient?
d. Suppose a random variable X is Binomial(60,.8). What is the probability X is less than 45? Include any
necessary checks for using approximations to distributions.