Recommended HW 2 F23
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School
University of Michigan *
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Course
250
Subject
Chemistry
Date
Dec 6, 2023
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36
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Question 1 : Comparing Histograms: Measures of Center and Variability
Let's look at some histograms for quantitative variables and apply the concepts from lectures 4
and 5.
1.a)
Which histogram has the smallest mean value?
( ) Histogram A
( ) Histogram B
( ) Histogram C
1.b)
Which histogram has the largest standard deviation value?
( ) Histogram A
( ) Histogram B
( ) Histogram C
1.c)
When comparing the standard deviations of Histograms A and B, which of the following(s)
apply?
( ) Histogram A and B have similar standard deviation values
( ) The standard deviation value for Histogram A is larger than that of Histogram B
( ) The standard deviation value for Histogram B is larger than that of Histogram A
( ) The standard deviation value for Histograms A and B is roughly 5 units.
Question 2 : Exam 1 vs. Exam 2
The mean score for all students taking Exam 1 was 70 and the standard deviation was 5. The
mean score for all students taking Exam 2 was 50 and the standard deviation was 10.
2.a) A student received a score of 85 on Exam 1. What is the student's standard score?
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2.b) A student received a standard score of -1.7 on Exam 2. What is the student's actual score
on Exam 2?
2.c) Gloria took Exam 1 and Abel took Exam 2. Gloria got 77.5 points on Exam 1 and Abel got 65
points on Exam 2. Who performed better, relative to their classmates, Gloria or Abel?
Question 3: SAT Scores
Victor, a recent high school student, took the SAT exam in 2019 and got a 600 in all three
components (Critical Reading, Math, and Writing). He was interested in how well he did
compare to the rest of his peers. The table below shows the summary statistics for all students
in 2019.
Which of Victor's three scores is the least unusual relative to his peers? In which component did
Victor perform best relative to his peers? Explain. This a friendly reminder to please show all
supporting work.
Question 4 : Screen Time: Missing Standard Deviation (Measure of Variability)
Data analysts at Apple wanted to learn about screen time habits for iPhone 14 Pro Max users. To
do so, a large random sample of iPhone 14 Pro Max users was selected, and for each user, the
average daily screen time was recorded (in minutes). Based on the data, the distribution of
average daily screen time for iPhone 14 Pro Max users is symmetric and unimodal (a
bell-shaped distribution) with a mean of 372 minutes (6 hours and
12 minutes). Furthermore, the smallest reported value was 78 minutes (1 hour and 18 minutes),
and the largest reported value was 666 minutes (11 hours and 6 minutes).
The analysts forgot to provide the standard deviation when providing the final report. Based on
the information provided, provide an approximate value for missing standard deviation. Include
a complete explanation of how you obtained the missing standard deviation value.
Question 5 : How the World Sleeps
Smartphones help researchers uncover how the world sleeps ~ A study, led by U-M researchers,
used a free smartphone app to gather robust sleep data from people in many nations. They
examined how gender, primary light source (some outdoor sunlight versus mostly indoor light),
and home country (U.S., Japan, Netherlands, etc.) affect the amount of sleep (in hours) people
get. For example, the average amount of sleep for residents of Japan was 7.5 hours, while the
average amount of sleep for those in the Netherlands was 8.25 hours.
a)
For this study, the variable
amount of sleep
is: (select all that apply).
( ) a categorical variable
( ) a quantitative variable
( ) an explanatory variable
( ) a response variable
b)
These boxplots compare the amount of sleep for the 1200 adults who spent most of
their time in indoor light and the 700 adults who spent some of their time outdoors.
Based on these boxplots, is the following statement TRUE or FALSE?
"The distribution of the amount of sleep for adults who spend most of their time in indoor
light is symmetric."
( ) True
( ) False
c)
These boxplots compare the amount of sleep for the 1200 adults who spent most of
their time in indoor light and the 700 adults who spent some of their time outdoors.
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Based on these boxplots, which value best completes this statement?
"For adults who spend most of their time in indoor light, 75% get at least _____________ hours
of sleep."
( ) 6.5
( ) 7
( ) 7.5
( ) 8
d)
These boxplots compare the amount of sleep for the 1200 adults who spent most of
their time in indoor light and the 700 adults who spent some of their time outdoors.
Based on these boxplots, which value best completes this statement?
"The most amount of sleep that an adult who spends some of their time outdoors in the
sunlight each day was _________ hours."
( ) 6.5
( ) 7
( ) 7.5
( ) 8
( ) 9
( ) 10
e)
These boxplots compare the amount of sleep for the 1200 adults who spent most of
their time in indoor light and the 700 adults who spent some of their time outdoors.
Based on these boxplots, which value best answers the question: "For this study, how many of
the adults who spend some of their time outdoors get between 7 and 8 hours of sleep?"
( ) 50
( ) 175
( ) 350
( ) 600
Question 6 : Great Danes
A certain breed of dog, Great Danes, are one of the tallest breeds of dogs. The heights of Great
Danes follow a normal distribution with a mean of 32 inches and a standard deviation of 2.5
inches.
a.
What is the minimum height a Great Dane needs to be so that it is taller than at least
75% of the other Great Danes?
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b.
A particular Great Dane measures 35 inches tall. About what percent of Great Danes are
taller than that dog? Show your work. Your work may include an upload of a sketch or
graph if you wish.
Question 7 : S&P 500: Performance of 500 large companies
The Standard and Poor's 500, or simply the S&P 500, is a stock market index tracking the
performance of 500 large companies listed on stock exchanges in the United States. On the
other hand, the SPY-ETF is an exchange-traded fund (ETF) created to provide individuals with an
investment vehicle to invest in the stock market. SPY-ETF tracks the S&P 500 index by holding a
portfolio of stocks in companies in the S&P 500, producing returns on investments roughly in
line with the S&P 500 index.
Assume that the S&P 500 index level follows a normal distribution with a mean index level of
3,915 points and a standard deviation of 257 points. Suppose you are an analyst working in a
Hedge Fund, and your manager provides the following investment rules:
➢
If the reported S&P 500 index level falls in the top 20% of the point distribution, sell 15%
of the assets in the SPY-ETF.
➢
If the reported S&P 500 index level is below 3,745 points, buy additional shares of the
SPY-ETF.
7.a) What is the probability you will buy additional shares of the SPY-ETF? In other words, what
is the 1 point(s) probability that the S&P 500 index level falls below 3,745 points? Please round
your answer to four decimal places.
( ) 0.7458
( ) 0.6832
( ) 0.6615
( ) 0.3168
( ) 0.2542
( ) -0.6615
7.b) What is the minimum index level that the S&P 500 should reach in order to sell 15% of the
assets in the SPY-ETF? Please round your answer to two decimal places. Here are the investment
rules provided by the manager:
➢
If the reported S&P 500 index level falls in the top 20% of the point distribution, sell
15% of the assets in the SPY-ETF.
➢
If the reported S&P 500 index level is below 3,745 points, buy additional shares of the
SPY-ETF.
( ) 3,698.70 points
( ) 4,097.82 points
( ) 3,986.78 points
( ) 4,131.29 points
( ) 3,843.22 points
Question 8 : Chemistry Exam Scores
Suppose a professor teaches a general chemistry class and an organic chemistry class. The
questions that follow are about the exam scores for these two classes.
a.
Suppose that
general chemistry
exam scores are normally distributed with a mean of 66
points and 2 point(s) a standard deviation of 12 points.
Find the value that completes the following statement:
7% of students have a score of _________ points or higher on the general chemistry
exam.
Be sure that you provide some work/logic for how you arrived at your answer.
b.
Suppose that
organic chemistry
exam scores are also normally distributed with a mean
of 80 points.
William got his organic chemistry score back. He decides to ask the professor how well
he did in the exam compared to other students who took organic chemistry. The
professor responds that 13% of students who took the organic chemistry exam got a
score of 92 points or above.
Based on the information provided, what is the
approximate
value of the standard
deviation for exam scores on the organic chemistry exam?
Question 9 : 25-yard Freestyle
According to the USA Swimming association, the distribution of finishing time, in seconds, for a
25-yard Freestyle competition for girls 8 years and under is approximately bell-shaped with a
mean of 41 seconds and a standard deviation of 5 seconds.
a.
Which of the following graphical techniques would be better to help assess if a
“bell-shaped” distribution is a reasonable model for the underlying population of such
finishing times?
( ) Bar Chart
( ) Histogram
( ) Pie Chart
( ) Boxplot
b.
Natalie is a 7 year old girl that participated in a 25-yard Freestyle competition and her
finishing time is) 36 seconds. Based on the stated model,
approximately
, what
percentage of the 25-yard Freestyle finishing times for girls 8 years and under are
expected to be slower than Natalie?
( ) 95%
( ) 68%
( ) 32%
( ) 16%
( ) 84%
( ) 2.5%
( ) 5%
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c.
Sarah is an 8 year old girl that participated in a 25-yard Freestyle competition and her
finishing time was 48.5 seconds. Which of the following is a correct statement about
Sarah's finishing time?
( ) Sarah's finishing time in a 25-yard Freestyle competition is 7.5 standard deviations below the
mean finishing time.
( ) Sarah's finishing time in a 25-yard Freestyle competition is 7.5 standard deviations above the
mean finishing time.
( ) Sarah's finishing time in a 25-yard Freestyle competition is 1.5 standard deviations below the
mean finishing time.
( ) Sarah's finishing time in a 25-yard Freestyle competition is 1.5 standard deviations above the
mean finishing time.
d.
Find the value to complete the following statement: The fastest 2.5% of female
swimmers 8 years and under, will complete a 25-yard Freestyle competition in
at most
_________ seconds, approximately.
( ) 26
( ) 31
( ) 36
( ) 41
( ) 46
( ) 51
( ) 56
Question 10: Disarming fish ~
The ‘stickleback’ is a species of coastal fish named for its
defensive armor, with a number of lateral bony ‘spike’ plates down both sides. This armor
reduces mortality from ocean fish and diving birds. In contrast, in lakes and streams, where
there are fewer predators, stickleback populations have reduced armor. Researchers
[1]
have
found that much of the difference in number of plates is caused by a single gene,
Ectodysplasin
.
Fish with two copies of the oceanic gene,
MM
, had many plates, whereas fish with two copies of
the freshwater gene,
mm
, had few plates. Fish inheriting both copies
Mm
had a wide range of
plate numbers. Plates are counted as the total number down the left and right sides of the fish.
The total number of fish sampled for each genotype are: 82 (
MM
); 174 (
Mm
); and 88 (
mm
).
Use this plot to make decisions about the relationship between Quantity A and Quantity B for
each row in the table. In each case, select the most appropriate statement from the following
choices (you may use each choice more than once or not at all).
a.
Quantity A is greater
b.
Quantity B is greater
c.
The quantities are the same
d.
The relationship cannot be determined without more information
Statement
a, b, c, or d
Quantity A
Quantity B
The
75
th
percentile for the Mm
group.
The
25
th
percentile for the MM
group.
The
z-score
of a ‘MM’
stickleback with 60 lateral
plates.
The
z-score
of a ‘Mm’
stickleback with 60 lateral
plates.
The
range
of lateral plate
counts for the mm group.
The
IQR
of lateral plate counts
for the Mm group.
The mean number of plates of
‘Mm’ stickleback fish.
The median number of plates
of ‘Mm’ stickleback fish.
Solutions
Question 1 : Comparing Histograms: Measures of Center and Variability
Let's look at some histograms for quantitative variables and apply the concepts from lectures 4
and 5.
1.a)
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Which histogram has the smallest mean value?
(X) Histogram A
( ) Histogram B
( ) Histogram C
1.b)
Which histogram has the largest standard deviation value?
( ) Histogram A
( ) Histogram B
(X) Histogram C
1.c)
When comparing the standard deviations of Histograms A and B, which of the following(s)
apply?
(X) Histogram A and B have similar standard deviation values
( ) The standard deviation value for Histogram A is larger than that of Histogram B
( ) The standard deviation value for Histogram B is larger than that of Histogram A
(X) The standard deviation value for Histograms A and B is roughly 5 units.
Question 2 : Exam 1 vs. Exam 2
The mean score for all students taking Exam 1 was 70 and the standard deviation was 5. The
mean score for all students taking Exam 2 was 50 and the standard deviation was 10.
2.a) A student received a score of 85 on Exam 1. What is the student's standard score?
Answer:
The student's standard score is (85-70)/5 = 3.
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2.b) A student received a standard score of -1.7 on Exam 2. What is the student's actual score
on Exam 2?
Answer:
The student's actual score is 50 - 1.7*10 = 33 points.
2.c) Gloria took Exam 1 and Abel took Exam 2. Gloria got 77.5 points on Exam 1 and Abel got 65
points on Exam 2. Who performed better, relative to their classmates, Gloria or Abel?
Question 3: SAT Scores
Victor, a recent high school student, took the SAT exam in 2019 and got a 600 in all three
components (Critical Reading, Math, and Writing). He was interested in how well he did
compare to the rest of his peers. The table below shows the summary statistics for all students
in 2019.
Which of Victor's three scores is the least unusual relative to his peers? In which component did
Victor perform best relative to his peers? Explain. This a friendly reminder to please show all
supporting work.
Question 4 : Screen Time: Missing Standard Deviation (Measure of Variability)
Data analysts at Apple wanted to learn about screen time habits for iPhone 14 Pro Max users. To
do so, a large random sample of iPhone 14 Pro Max users was selected, and for each user, the
average daily screen time was recorded (in minutes). Based on the data, the distribution of
average daily screen time for iPhone 14 Pro Max users is symmetric and unimodal (a
bell-shaped distribution) with a mean of 372 minutes (6 hours and
12 minutes). Furthermore, the smallest reported value was 78 minutes (1 hour and 18 minutes),
and the largest reported value was 666 minutes (11 hours and 6 minutes).
The analysts forgot to provide the standard deviation when providing the final report. Based on
the information provided, provide an approximate value for missing standard deviation. Include
a complete explanation of how you obtained the missing standard deviation value.
Answer:
The key here is to use the empirical rule: Based on the empirical rule, 99.7% of the observations
are within 3 standard deviations of the mean. In other words, the overall range should consist of
about
6 standard deviations.
The background provides a mean of 372 minutes, the minimum is 78 minutes, and the
maximum is 666 minutes. We find that the range is 666 - 78 = 588. Dividing by 6 for the 6
standard deviations, we get 588/6 = 98.
Equivalently, we can also calculate the difference between one end and the mean and divide it
by 3 standard deviations. For instance, (max - mean)/3 = (666 - 372)/3 = 294/3 = 98 and (mean -
min)/3 = (372-78) = 294/3 = 98.
Question 5 : How the World Sleeps
Smartphones help researchers uncover how the world sleeps ~ A study, led by U-M researchers,
used a free smartphone app to gather robust sleep data from people in many nations. They
examined how gender, primary light source (some outdoor sunlight versus mostly indoor light),
and home country (U.S., Japan, Netherlands, etc.) affect the amount of sleep (in hours) people
get. For example, the average amount of sleep for residents of Japan was 7.5 hours, while the
average amount of sleep for those in the Netherlands was 8.25 hours.
a)
For this study, the variable
amount of sleep
is: (select all that apply).
( ) a categorical variable
(X) a quantitative variable
( ) an explanatory variable
(X) a response variable
Explanation:
The statement: "examined how gender, primary light source (some outdoor sunlight versus
mostly indoor light), and home country (U.S., Japan, Netherlands, etc.) affect the amount of
sleep (in hours) people get" establishes that the amount of sleep is the primary outcome
variable of interest in the study. Because the amount of sleep was recorded in hours, it is
quantitative.
b)
The study found that women schedule more sleep than men, on average. However, it
was discovered 1 point(s) that the difference was most pronounced for middle-aged
adults, and somewhat negligible for seniors. Based on these results, the variable “age
group” would be called: (select one).
( ) a response variable
(X) a confounding variable
( ) a randomized variable
Explanation:
The statement: "examined how gender, primary light source (some outdoor sunlight versus
mostly indoor light), and home country (U.S., Japan, Netherlands, etc.) affect the amount of
sleep (in hours) people get" establishes that the age was neither an intended explanatory
variable nor a response variable. However, age group was recorded and upon looking further at
the results, found to be related to the response variable amount of sleep and also tied to the
explanatory variable gender. Thus age group is a confounding variable.
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c)
These boxplots compare the amount of sleep for the 1200 adults who spent most of
their time in indoor light and the 700 adults who spent some of their time outdoors.
Based on these boxplots, is the following statement TRUE or FALSE?
"The distribution of the amount of sleep for adults who spend most of their time in indoor
light is symmetric."
( ) True
(X) False
Explanation:
Boxplots do not show the underlying shape of the distribution well. Boxplots can hide gaps and
clusters too. We only know that the quartiles are approximately symmetric around the median
as well as the min and max, but we do not know how the values between these summaries are
distributed.
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d)
These boxplots compare the amount of sleep for the 1200 adults who spent most of
their time in indoor light and the 700 adults who spent some of their time outdoors.
Based on these boxplots, which value best completes this statement?
"For adults who spend most of their time in indoor light, 75% get at least _____________ hours
of sleep."
(X) 6.5
( ) 7
( ) 7.5
( ) 8
Explanation:
First, be sure to work with the boxplot for indoor light. Then since Q1 = 6.5 hours, about 25% of
the adults get 6.5 hours or less (at most 6.5 hours), so there are about 75% of the adults who
get at least 6.5 hours of sleep.
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e)
These boxplots compare the amount of sleep for the 1200 adults who spent most of
their time in indoor light and the 700 adults who spent some of their time outdoors.
Based on these boxplots, which value best completes this statement?
"The most amount of sleep that an adult who spends some of their time outdoors in the
sunlight each day was _________ hours."
( ) 6.5
( ) 7
( ) 7.5
( ) 8
( ) 9
(X) 10
Explanation:
First, be sure to work with the boxplot for outdoor light. Then remember that outliers are still
part of the data set so the largest value was 10 hours.
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f)
These boxplots compare the amount of sleep for the 1200 adults who spent most of
their time in indoor light and the 700 adults who spent some of their time outdoors.
Based on these boxplots, which value best answers the question: "For this study, how many of
the adults who spend some of their time outdoors get between 7 and 8 hours of sleep?"
( ) 50
( ) 175
(X) 350
( ) 600
Explanation:
Since there were 700 adults in the outdoor group, and about 50% of them will have between
Q1=7 hours and Q3=8 hours of sleep, we have 50% of 700 for 350 adults in the study.
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Question 6 : Great Danes
A certain breed of dog, Great Danes, are one of the tallest breeds of dogs. The heights of Great
Danes follow a normal distribution with a mean of 32 inches and a standard deviation of 2.5
inches.
a.
What is the minimum height a Great Dane needs to be so that it is taller than at least
75% of the other Great Danes?
b.
A particular Great Dane measures 35 inches tall. About what percent of Great Danes are taller
than that dog? Show your work. Your work may include an upload of a sketch or graph if you
wish.
OR
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Question 7 : S&P 500: Performance of 500 large companies
The Standard and Poor's 500, or simply the S&P 500, is a stock market index tracking the
performance of 500 large companies listed on stock exchanges in the United States. On the
other hand, the SPY-ETF is an exchange-traded fund (ETF) created to provide individuals with an
investment vehicle to invest in the stock market. SPY-ETF tracks the S&P 500 index by holding a
portfolio of stocks in companies in the S&P 500, producing returns on investments roughly in
line with the S&P 500 index.
Assume that the S&P 500 index level follows a normal distribution with a mean index level of
3,915 points and a standard deviation of 257 points. Suppose you are an analyst working in a
Hedge Fund, and your manager provides the following investment rules:
➢
If the reported S&P 500 index level falls in the top 20% of the point distribution, sell 15%
of the assets in the SPY-ETF.
➢
If the reported S&P 500 index level is below 3,745 points, buy additional shares of the
SPY-ETF.
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7.a) What is the probability you will buy additional shares of the SPY-ETF? In other words, what
is the 1 point(s) probability that the S&P 500 index level falls below 3,745 points? Please round
your answer to four decimal places.
( ) 0.7458
( ) 0.6832
( ) 0.6615
( ) 0.3168
(X) 0.2542
( ) -0.6615
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7.b) What is the minimum index level that the S&P 500 should reach in order to sell 15% of the
assets in the SPY-ETF? Please round your answer to two decimal places. Here are the investment
rules provided by the manager:
➢
If the reported S&P 500 index level falls in the top 20% of the point distribution, sell
15% of the assets in the SPY-ETF.
➢
If the reported S&P 500 index level is below 3,745 points, buy additional shares of the
SPY-ETF.
( ) 3,698.70 points
( ) 4,097.82 points
( ) 3,986.78 points
(X) 4,131.29 points
( ) 3,843.22 points
Explanation:
We want the top 20%, which corresponds to the 80th percentile, and a z-score of 0.84. We use
the formula for z-score to get our minimum index level:
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Question 8 : Chemistry Exam Scores
Suppose a professor teaches a general chemistry class and an organic chemistry class. The
questions that follow are about the exam scores for these two classes.
a.
Suppose that
general chemistry
exam scores are normally distributed with a mean of 66
points and 2 point(s) a standard deviation of 12 points.
Find the value that completes the following statement:
7% of students have a score of _________ points or higher on the general chemistry
exam.
Be sure that you provide some work/logic for how you arrived at your answer.
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b.
Suppose that
organic chemistry
exam scores are also normally distributed with a mean
of 80 points.
William got his organic chemistry score back. He decides to ask the professor how well
he did in the exam compared to other students who took organic chemistry. The
professor responds that 13% of students who took the organic chemistry exam got a
score of 92 points or above.
Based on the information provided, what is the
approximate
value of the standard
deviation for exam scores on the organic chemistry exam?
Explanation:
Based on the information provided, we are told that the distribution of
organic chemistry
exam
scores is also normally distributed with a mean of 80 points. Furthermore, we are told that 13%
of students scored 92 points or above.
Question 9 : 50-yard Freestyle
According to the USA Swimming association, the distribution of finishing time, in seconds, for a
50-yard Freestyle competition for girls 8 years and under is approximately bell-shaped with a
mean of 41 seconds and a standard deviation of 5 seconds.
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a.
Which of the following graphical techniques would be better to help assess if a
“bell-shaped” distribution is a reasonable model for the underlying population of such
finishing times?
( ) Bar Chart
(X) Histogram
( ) Pie Chart
( ) Boxplot
Explanation:
One appropriate graph for assessing shape for a quantitative variable would be a
histogram.
A boxplot cannot be used to confirm if a bell-shaped distribution is reasonable.
Finally, graphs such as bar graphs or pie charts would not be correct here as they are used for
categorical data.
b.
Natalie is a 7 year old girl that participated in a 50-yard Freestyle competition and her
finishing time is) 36 seconds. Based on the stated model,
approximately
, what
percentage of the 50-yard Freestyle finishing times for girls 8 years and under are
expected to be slower than Natalie?
( ) 95%
( ) 68%
( ) 32%
( ) 16%
(X) 84%
( ) 2.5%
( ) 5%
Explanation:
Since the model is bell-shaped, by the empirical rule, we would expect about 68% of the
observations to be within one standard deviation from the mean, that is, between 36 seconds
and 46 seconds. That would leave about 32% of the observations to be in the two 'tails' (outside
of the 36 to 46 seconds range). Thus, about 16% of the finishing times would be to the left of 36
seconds and 84% of finishing times would be right of 36 seconds. We are looking for the
percentage of finishing times that are
slower
than Natalie, hence have a higher finishing time
than 36 seconds, hence 84%.
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Using the R Shiny App we have:
c.
Sarah is an 8 year old girl that participated in a 50-yard Freestyle competition and her
finishing time was 48.5 seconds. Which of the following is a correct statement about
Sarah's finishing time?
( ) Sarah's finishing time in a 50-yard Freestyle competition is 7.5 standard deviations below the
mean finishing time.
( ) Sarah's finishing time in a 50-yard Freestyle competition is 7.5 standard deviations above the
mean finishing time.
( ) Sarah's finishing time in a 50-yard Freestyle competition is 1.5 standard deviations below the
mean finishing time.
(X) Sarah's finishing time in a 50-yard Freestyle competition is 1.5 standard deviations above
the mean finishing time.
Explanation: We need to compute the standard score here.
Standard score = (obs- mean)/ s.d. = (48.5 – 41) / 5 = 1.5 So Sarah's finishing time for the
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50-yard Freestyle competition is 1.5 standard deviations ABOVE the mean finishing time of 41
seconds.
d.
Find the value to complete the following statement: The fastest 2.5% of female
swimmers 8 years and under, will complete a 50-yard Freestyle competition in
at most
_________ seconds, approximately.
( ) 26
(X) 31
( ) 36
( ) 41
( ) 46
( ) 51
( ) 56
Explanation:
Since the model is bell-shaped, by the empirical rule,we would expect about 95% of the
observations to be within two standard deviations from the mean, that is, between 31 seconds
and 51 seconds. That would leave about 5% of the observations to be in the two 'tails' (outside
of the 31 to 51 seconds range). Thus, about 2.5% of the observations are below 31 seconds and
2.5% of the observations are above 51 seconds. We are looking for the fastest 2.5% hence we
want the lower tail, 31 seconds and below.
Using the z table we could find the lower 2.5th percentile as z = -1.96 and solving for the value
of x we have: -1.96 = (x - 41)/5 --> x = 41 - 1.96(5) = 31.2 seconds.
Question 10: Disarming fish ~
The ‘stickleback’ is a species of coastal fish named for its
defensive armor, with a number of lateral bony ‘spike’ plates down both sides. This armor
reduces mortality from ocean fish and diving birds. In contrast, in lakes and streams, where
there are fewer predators, stickleback populations have reduced armor. Researchers
[1]
have
found that much of the difference in number of plates is caused by a single gene,
Ectodysplasin
.
Fish with two copies of the oceanic gene,
MM
, had many plates, whereas fish with two copies of
the freshwater gene,
mm
, had few plates. Fish inheriting both copies
Mm
had a wide range of
plate numbers. Plates are counted as the total number down the left and right sides of the fish.
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The total number of fish sampled for each genotype are: 82 (
MM
); 174 (
Mm
); and 88 (
mm
).
Use this plot to make decisions about the relationship between Quantity A and Quantity B for
each row in the table. In each case, select the most appropriate statement from the following
choices (you may use each choice more than once or not at all).
a.
Quantity A is greater
b.
Quantity B is greater
c.
The quantities are the same
d.
The relationship cannot be determined without more information
Statement
a, b, c, or d
Quantity A
Quantity B
C
The
75
th
percentile for the Mm
group.
The value is about 62.5 (the
upper bound of the box)
The
25
th
percentile for the MM
group.
The value is about 62.5 (the
lower bound of the box)
A
The
z-score
of a ‘MM’
stickleback with 60 lateral
plates.
The observation is going to be
above the mean, a +z-score
The
z-score
of a ‘Mm’
stickleback with 60 lateral
plates.
The observation is going to be
below the mean, a -z-score
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A
The
range
of lateral plate
counts for the mm group.
The range is about 30 lateral
plates (from 6 to about 36)
The
IQR
of lateral plate counts
for the Mm group.
The IQR is about 20 lateral
plates (from about 41 to 62)
B
The mean number of plates of
‘Mm’ stickleback fish.
The mean is less than median
due to the long lower whisker
and the location of the
median value close to the
75th percentile.
The median number of plates
of ‘Mm’ stickleback fish.
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