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SOLUTIONS
Stat 250.3 First Midterm Exam
Wednesday, October 15, 2003
Part I: (100 points) Written Problems: Show ALL work, calculations and formulas used. Partial
credit will be awarded for using the correct procedures.
1. Joe is interested in conducting a study to determine if exposure to the sun (# hours per week in the
sun) is related to skin cancer. [18]
(a) What are the two variables of interest? What type is each of these variables, quantitative,
categorical or ordinal? [8]


Numbers of hours per week in the sun – Quantitative
Having or not a skin cancer - Categorical
(b) Which one is the explanatory variable and which one is the response variable? [6]


Numbers of hours per week in the sun – Explanatory
Having or not a skin cancer - Response
(c) Ethically, what type of study would I have to do to determine if there is a relationship
between the two variables? [4]
Observational study
2. Suppose that the probability of having a girl is .45. A couple has decided to have 5 children. Let
X = the number of girls out of the five children. Below are a partial PDF and CDF for X. [22]
(a) The notation P(X ≤ x) refers to which of the distribution functions? [4]
CDF
(b) Complete the table below. [12]
K
0
1
2
3
4
5
P(X=k)
0.050
0.206
0.337
0.276
0.113
0.018
P(X≤k)
0.050
0.256
0.593
0.869
0.982
1.000
(c) What is the probability that the couple will have an odd number of girls? [6]
0.206+0.276+0.018 = 0.5
3. A basketball player has 60% probability of success on a free throw. Suppose that in a game he
attempted 12 free throws, and let X be the number of free throws he made. We can assume that
the result in each of the free throws is independent from the others. [30]
(a) What is the distribution of X? (Include the parameter values necessary to fully describe its
distribution!) [10]
X is binomial random variable n = 12 and p = 0.6.
(c) Find the probability that he made at least five free throws. [10]
P( X  5)  1  P( X  4)  1  0.0573  0.9427
(c) Find the mean and the variance of X. [10]
μ = E(X) = np = 12( .6) = 7.2
σ2 = Var(X) = np(1-p) = 12(.6)(.4) = 2.88
4. Suppose the number of calories PSU students consume in a day is normally distributed with mean
2000 and standard deviation 300. [30]
a) What is the probability that a randomly selected individual consumed between 1800 and
2100 calories yesterday? [10]
P(1800< X < 2100) = P( -200/300 < Z < 100/300) = P(-0.67 < Z < 0.33)
= P (Z < 0.33) – P(Z<-0.67) = 0.6293 – 0.2514 = 0.3779
b) About 95% of PSU students have a daily caloric intake between what two values? [10]
μ ±2σ = 2000 ± 600, i.e from 1400 to 2600 calories.
c) What is the 90th percentile for the amount of calories consumed in a day by a PSU
student? (i.e. what is the amount of calories, that 90% of the PSU students consume less
calories than that in a day?) [10]
P (Z < z)=.90 ==> z = 12.8, thus x = 300 (1.28) + 2000 = 2384
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Midterm Exam Part II: (100 points) Multiple Choice Problems
SOLUTIONS: A-B-C-A-A B-C-D-B-C D-A-A-D-C B-C-C-A-B
1. Which of these is a quantitative variable?
A. The length of your right foot.
B. The month you were born.
C. The political party you belong to.
D. The newspaper you prefer to read.
2. Researchers were interested in the relationship between whether students attended a public or private high
school and involvement in activities at college. Fifty public school students and 50 private school students
were randomly chosen to participate in the study. Which of the following statements is true?
A. Type of school and involvement in activities are both response variables.
B. Type of school is an explanatory variable and involvement in activities is a response variable.
C. Type of school is a response variable and involvement in activities is an explanatory variable.
D. There is not enough information given about the study to classify the variables.
3. Which of the following can be used to summarize the two random variables in question 2?
A. Stem and leaf plot
B. Dot plot
C. Two-way table
D. Pie Chart
__________________________________________________________________________
Questions 4 and 5: A sample of students was asked if they believe that it is right or wrong to have sex before
marriage. Below is a table summarizing the responses for males and females in the sample.
Rows: Gender
Columns: SexB4Mar
Right
Wrong
All
122
43
165
96
20
116
63
Count
281
female
male
All
218
Cell Contents –
4. What percent of males in the sample believe it is wrong to have sex before marriage?
A. 0.17
B. 0.21
C. 0.07
D. 0.83
5. What percent of all those surveyed are females who believe it is wrong to have sex before marriage?
A. 0.15
B. 0.68
C. 0.26
D. 0.59
6. The shape of the data below is described as __________________.
B. left-skewed C. right-skewed
D. normal
10
Frequency
A. bell-shaped
5
0
50
55
60
65
70
75
80
85
90
scores
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95
100
________________________________________________________________________________
Questions 7 and 8: The following are the responses of 11 people when asked how much change (in cents)
they had in their wallet.
12
14
15
16
22
26
46
54
61
64
85
7. What is the median for this data?
A. 22
B. 36
C. 26
D. 24
8. Using the data in the previous question, what is the IQR?
A. 6
B. 35
C.73
D. 46
_________________________________________________________________________________
9. A data value that is inconsistent with the bulk of the data is a(n):
A. minimum
B. outlier
C. maximum
D. placebo effect
10. The third quartile is:
A. 75
B. the data value that is .75 standard deviations from the mean
C. the data value below which 75% of the values fall
D. the data value below which 25% of the values fall
11. A researcher wants to determine the effect of a new drug on growth. He selects 40 adolescents for the
study and randomly divides all subjects into two groups. He will treat one of the groups using the new drug
and the other group placebo. This is an example of a(n):
A. Observational study
B. Matched pair design
C. Latin square design
D. Randomized design
12. In reference to a five-number summary, fifty percent of the data values fall between what two numbers?
A. first quartile and third quartile
B. third quartile and maximum
C. first quartile and maximum
D. minimum and maximum
13. A variable that affects the response variable and is related to the explanatory variable is known as:
A. confounding
B. skewing
C. discrete
D. random
14. Which of the following is a discrete random variable?
A.
B.
C.
D.
The time in seconds that an elephant can hold its breath underwater.
The height of an ant on an elephant in Zimbabwe.
The weight of an adult, ant free, female elephant in Zimbabwe.
The number of ants on an elephant in Zimbabwe.
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15. The payoff (X) for a lottery game has the following probability distribution.
k
P(X=k)
$0
0.8
$5
0.2
What is the expected value of X= payoff?
A. $0
B. $0.50
C. $1.00
D. $2.50
16. A birth is selected at random. Define events A=baby is a boy and B=the mother had the flu during her
pregnancy. The events A and B are
A. Disjoint but not independent.
B. Independent but not disjoint.
C. Disjoint and independent.
D. Neither disjoint nor independent.
17. What is the probability that Z is between 1 and 1, P(1  Z  1)? (Z is a standard normal random variable.)
A. 0.1587
B. 0.3174
C. 0.6826
D. 0.8413
18. The time taken for a computer to boot up, X, follows a normal distribution with mean 30 seconds and standard
deviation 5 seconds. What is the standardized score (z-score) for a boot up of time x =35 seconds?
A. 2.0
B. 0.0
C. 1.0
D. 2.0
_______________________________________________________________________________________
Questions 19 and 20: Suppose that P (  Z   )   , where Z is a standard normal random variable, 
is a positive number and  is a number between 0 and 1. (e.g for P(2  Z  2)  .95,   2 and
  .95 ). Find the following probabilities in terms of  : (it might be helpful to draw the standard normal
curve and draw the desired areas for each question.)
19. P(0  Z   )
A.

2
B.
1 
2
C. 1  
D. 2 
B.
1 
2
C. 1  
D. 2 
20. P ( Z   )
A.

2
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