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STAT 226
Supplemental Instruction
Practice Exam 1
Equation Review
Z-Score of a
Normal
Distribution
𝑍=
𝑥−𝜇
𝜎
Finding an
observation (x)
when given a ZScore
𝑥 = (𝑍)(𝜎) + 𝜇
Z- Score for
Sample
Means
(CLT)
𝑥−𝜇
𝑍= 𝜎
√𝑛
Mean
𝜇=
Standard Deviation
𝑥1 + 𝑥2 + 𝑥3
𝑛
(𝑥 − 𝜇)2 + (𝑥 − 𝜇)2
𝜎=
𝑛−1
Concept Review
1. How to describe a Histogram
a. Shape
b. Center (Mean or Median)
c. Variability (Standard Deviation or IQR)
d. Outliers
2. Central Limit Theorem: a large sample size from a population, that is not normal, says that the
mean of all samples will be approximately equal to the mean of the population.
a. Increase the sample size; increases precision, decreases bias, and decreases variability.
3. How to Interpret a Normal Quantile Plot
a. Do the data points follow close the line?
b. Are all of the points within the two ellipses?
i. Yes; Can be a normal distribution.
ii. No; Cannot be a normal distribution.
4. What does Pr(Z > 3) Mean?
a. “What is the probability that an observation has a Z-score greater than 3?”
5. Mean vs Median
Mean =Median
Mean < Median
Mean > Median
How to Find Z-Score/P-Value
Do I have a
population with
a Normal
Distribution?
Yes.
Can Find the Z-Score
and Probability of a
single observation.
No.
Is the distribution
Approximately Normal?
Yes.
Need a sample of at
least 15 to use the
Central Limit Theorem.
No
Need a sample of at
least 30 to use the
Central Limit Theorem.
STAT 226
Supplemental Instruction
Practice Exam 1
Multiple Choice Questions
1. The weight of every player on the Iowa State University football roster is recorded at the
start of the season. Suppose, when entering the weight of a 300-lb offensive lineman, the
assistant coach mistakenly enters 800 lb. Which of the following summary statistics of
the weight will be affected? (circle all that apply) a. Mean
d. Median
b. Interquartile Range
e. Mode
c. Standard Deviation
f. Range
2. Given that a population follows a N(85,3) distribution, the approximate value of the 84th
percentile is (circle one)
a. 88
c. 79
b. 82
d. 91
3. Which of the following are typically recorded as nominal variables?
I. Age
II. Gender
a. I and II
b. III and IV
III. Eye Color… ...IV. Grade Point Average (GPA)
c. I and IV
d. II and III
4. The height of male undergrads follows a Normal distribution, X~N(68,25). What range
contains approximately 95% of all male undergrads?
a. (43,93)
c. (63,73)
b. (18,118)
d. (58,78)
5. Suppose X~N(10,20) and if P(X<x) = 0.5 then x is equal to:
a. 20
c. 10
b. 0
d. None of the above
6. A questionnaire concerning satisfaction with the Financial Aid office on campus was
mailed to 45 students on a university campus. The 45 students in this survey are an
example of a
a. Population
c. Parameter
b. Sample
d. Statistic
7. The 50th percentile of a standard normal distribution is:
a. 0
c. 1
b. 0.5
d. Not enough information
8. A set of test scores follows a Normal distribution, X~N(80,36). Which of the following
statements is true?
a. P(X > 116) = 0.16
c. P(X > 85) = P(X < 85)
b. P(X < 80) = 1
d. P(74 < X < 86) = 0.68
STAT 226
Supplemental Instruction
Practice Exam 1
TRUE or FALSE: Circle either T or F.
...T/F: For a normal distribution, an observation with a z-score of -1.5 is less unusual than an
…observation with a z-score of 1. T/F: If a distribution is skewed right, the median should be larger than the mean. T/F: A statistic is a numerical summary of a population. T/F: Categorical variables take numerical values for which arithmetic operations such as
..adding and averaging make sense. T/F: As the sample size increases, the population standard deviation decreases.
T/F: The Center Limit Theorem can always be used if the sample size is larger than 15.
T/F: A Pareto chart is a bar chart where the categories are ordered alphabetically.
T/F: The mean and standard deviation are sensitive to outliers.
T/F: For any normally distributed random variable, we know that approximately 68% of the
..observations will be further than 1 standard deviation away from the mean.
T/F: An observation larger than the mean has a positive z-score.
T/F: The numbers 3, 4, 5 have the same standard deviation as 1003, 1004, 1005.
T/F: In a simple random sample, each set of n individuals has an equal chance of selection.
T/F: Categorical variables have a median.
T/F: The z-score tells us how many means we are away from the standard deviation.
T/F: The area under a density curve is always 1.
T/F: The empirical rule only applies to a standard normal distribution.
T/F: P(Z > c) = 1- P(Z<c)
STAT 226
Supplemental Instruction
Practice Exam 1
Short Answer
Apple’s New Product. The Apple Computer Company’s sales team is interested in launching a
new product in the United States. Before doing so, Apple wants to estimate the proportion of
college students around the U.S. that consider busying the new product. They pick one school at
random from each state and survey a simple random sample of 40 students from the selected
school. Their survey results indicated that 34% of the surveyed students would buy the product at
the proposed price. For this situation identify the following:
a) Population:
c) Sample:
b) Parameter:
d) Statistic:
Use the following data set;
1, 8, 8, 8, 9, 12, 14, 14, 15, 15, 17
a) Compute the Mean
b) Compute the Median
c) Compute the 5-Number Summary
d) Compute the Inter-Quartile Range
e) Name two methods that would be deemed appropriate for displaying this data. What
about the data allows you to reason that these are appropriate visual displays?
f) What would you consider to be appropriate measures of center and spread for this data?
Why?
STAT 226
Supplemental Instruction
Practice Exam 1
For this question, you must use the Empirical Rule. After finishing school, you decide to open
a sunglass-selling stand. Your stand is only pen during the summer. You know that based on
other sunglass selling stand in the area, that the overall summer sales of sunglasses for this area
is approximately Normally distributed with a mean of 500 sunglasses and a standard deviation of
65 sunglasses. Assuming your stands sales will follow the same distribution, answer the
following;
a) What is the probability that you sell more than 695 sunglasses?
b) Your goal for the first summer is to sell as many glasses as the middle 95% of all such
stands. Specify the range in which the number of glasses that you sell over the summer
has to fall inside.
c) You sell sunglasses for $15 each and would like your total sales for the summer to be at
least $6,525. What is the probability that you achieve this goal?
d) What price must you charge to sell enough sunglasses to fall at the 2.5th percentile but
still make at least $9,250?
5K Race. You and your friends are training for the annual Run of the Roses held in October in
Ames. Based on previous years’ data we can assume that the distribution of the time it takes to
complete the run is Normally distributed with a mean of 27 minutes and a stand deviation of 4
minutes in the 20 to 24 years’ age group. Use this information to answer the following questions.
a) What percent of participants cannot complete a 5 kilometer run in 30 minutes or less?
b) What percent of students run between 18 minutes and 27 minutes to complete the 5
kilometer run?
c) Your friend Max brags that he can finish the 5 kilometers faster than 90% of all
participants. How much time does he have to accomplish this feat?
STAT 226
Supplemental Instruction
Practice Exam 1
d) Your friend Fred runs for the ISU Track team and complete a 5 kilometer run in 15
minutes. What percent of students are faster than him?
e) A final friend, Dan, simply wants to finish the run ahead of the slowest 20% of all
participants. Assume it take him 34 minutes to complete the 5k on the day of the race.
Are 34 minutes fast enough to accomplish his goal?
New Camera. The Manufacturer of a new instant-picture camera claims that its product had “the
world’s fastest-developing color film by far.” Extensive laboratory testing has shown that the
distribution for the time it takes the new instant camera to begin to reveal the image after
shooting has a mean of 9.9 seconds and a standard deviation of 0.6 seconds. Suppose 50 of these
cameras are randomly selected from the production line and tested. The time until the image is
revealed is recorded for each.
a) Describe the sampling distribution of the sample mean, the time is takes the sample of 50
cameras to begin to reveal the image, i.e., give the mean, the standard deviation, and the
shape of the sampling distribution.
b) State the Theorem which is used for justification of the sampling distribution in (a).
Explicitly state the theorem, do not just give the name.
c) Find the probability that we will obtain a sample mean greater than 9.7 seconds.
d) Find the probability that we will obtain a sample mean between 9.8 seconds and 9.85
seconds.
e) What is the mean time until the image is first revealed for 50 sampled cameras such that
only 3% of every mean of sample size 50 will be lower?
f) Refer to part (a). Describe the changes in the sampling distribution of the sample mean if
the sample size is decreased from n=50 to n=20.
STAT 226
Supplemental Instruction
Practice Exam 1
STAT 226
Supplemental Instruction
Practice Exam 1