Expert answer:DATA Analysis
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data_analysis_assignment_3.docx
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STAT 250 Fall 2017 Data Analysis Assignment 3
Your submitted document should include the following items. Points will be deducted if the
following are not included. For an example of formatting, see the sample problem/solution
document posted on Blackboard.
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Type your Name and STAT 250 with your correct section number (e.g. STAT 250-xxx),
right justified on the top of page 1 of your document.
Type Data Analysis Assignment 3 centered on Page 1 under your name.
Number your pages across your entire solutions document.
Your document should include the ANSWERS ONLY with each answer labeled by its
corresponding number and subpart (there are four problems on this assignment). Keep
your solutions in order. You should not include the questions in your submitted
document.
Generate all requested graphs using StatCrunch. Graphs must be appropriately titled and
should refer to the context of the question. Graphical displays must include labels with
units if appropriate for each axis.
Upload your solutions document onto Blackboard as a Word or pdf document using the
link provided by your instructor.
Elements of good technical writing:
Use complete and coherent sentences to answer the questions.
Again, graphs must be appropriately titled and should refer to the context of the question.
Again, graphical displays must include labels with units if appropriate for each axis.
Units should always be included when referring to numerical values.
When making a comparison you must use comparative language, such as “greater than”, “less
than”, or “about the same as.”
Ensure that all graphs and tables appear on one page and are not split across two pages.
Type all mathematical calculations when directed to compute an answer ‘by-hand.’ Pictures of
actual handwritten work is not accepted.
When writing mathematical expressions into your document you may use either an equation
editor or common shortcuts such as:
x can be written as sqrt(x), p̂ can be written as p-hat, x
can be written as x-bar.
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Problem 1: STAT 250 Final Exam Scores
The data set “STAT 250 Final Exam Scores” contains a random sample of 385 STAT 250
students’ final exam scores (maximum of 100) collected over the past two years.
a) Use StatCrunch to construct an appropriately titled and labeled relative frequency
histogram of the final exam scores stored in the “Scores” variable. Copy your histogram
into your document. What is the shape of this distribution? Answer this question in one
complete sentence.
b) Now overlay your histogram with a Normal curve and add a vertical line at the mean.
(This can be done by going to Options Edit in the top left corner of your graph. Inside
the histogram graph box, look for Display Options. Next to “Overlay distrib.:” click the
arrow next to the word optional and select Normal. Then, check the box next to mean
under the word Markers.) Copy and paste this histogram into your document. Do you
think it is reasonable to use the normal model in this case? Answer this question in one
complete sentence.
c) Calculate the sample size, the mean, and the standard deviation of the “Scores” variable
using StatCrunch. (Select Stat Summary Stats Columns.) Copy and paste this
table into your document.
For parts (d) – (f), assume that the distribution of all final exam scores in the population is
Normal with the mean and standard deviation found in Part (c).
d) Calculate the probability that a randomly selected student scored above 88 on the final
exam. Draw a picture, shade area representing the desired probability, standardize, and
use the Standard Normal Table (Table 2 in your text) to obtain this probability. Please
take a picture of your hand drawn sketch and upload it to your Word document (if you do
not have this technology, you may use any other method (i.e. Microsoft paint) to sketch
the image). You must type the rest of your “by hand” work to earn full credit. Then,
once you obtain your answer, write one sentence to explain what the probability means in
context of the question. Finally, verify your answer using the StatCrunch Normal
calculator. Copy that image into your document as well.
e) Use StatCrunch only to calculate the probability that a randomly selected student scored
between 80 and 90. Copy the image from StatCrunch into your document. Then, once
you obtain your answer, write one sentence to explain what the probability means in
context of the question.
f) Suppose the professor decided that the bottom 14% would receive a letter grade of F on
the final exam. Determine the maximum score (or cutoff score) for which a student
would receive an F on the final exam. Draw a picture, shade area, standardize, and use
Table 2 to obtain this probability. Please take a picture of your hand drawn sketch and
upload it to your Word document (if you do not have this technology, you may use any
other method (i.e. Microsoft paint) to sketch the image). You must type the rest of your
“by hand” work to earn full credit. Then, once you obtain your answer, write one
sentence to explain what the probability means in context of the question. Finally, verify
your answer using the StatCrunch Normal calculator. Copy that image into your
document as well.
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Problem 2: Earning an A
An administrator takes a random sample of 21 students from one section STAT 250. The
random variable is defined as the number of students out of 21 that earn an A in STAT 250.
Assume that it is known that probability of earning an A in STAT 250 is 12% and assume that
the students’ grades are independent. Answer the following questions.
a) Check if this situation fits the binomial setting. Write four complete sentences addressing
each requirement in one sentence each.
b) Assuming it does, build the probability distribution in table form in StatCrunch. There
are two ways to do this. You may use Data Compute Expression and choose the
function dbinom. This method relies on you entering the values of the random variable in
the first column of your data table. The other way to do this is to use the binomial
calculator and calculate the probability of each of the values of the random variable from
X = 0 to X = 9. Since the probabilities will be very close to 0 after about X = 9, you
should cut off the table after X = 9. You may present this table horizontally or vertically.
c) Calculate the probability that exactly three students earned an A using the StatCrunch
binomial calculator. Copy this image from StatCrunch into your document. Then, once
you obtain your answer, write one sentence to explain what the probability means in
context of the question.
d) Calculate the probability that at least one student earned an A. Again, provide a
StatCrunch binomial calculator graph to display your answer. Then, once you obtain
your answer, write one sentence to explain what the probability means in context of the
question.
e) Calculate the probability that between 6 and 9 students (inclusive) earned an A. Show
your work using the probability distribution you built in part (b) to answer this question.
Then, verify it with a StatCrunch binomial calculator graph and include this image in
your document as well. Finally, once you obtain your answer, write one sentence to
explain what the probability means in context of the question.
f) Calculate the mean and standard deviation of this probability distribution. Show your
work using the binomial mean and standard deviation formulas and provide your answers
in your document. (No need to use StatCrunch for this part).
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Problem 3: Building a Sampling Distribution
Here we will use the Sampling Distribution applet to investigate sampling distributions of a
proportion of students who earned a B on their STAT 250 final exam. Historically, it is known
that 17.5% of students earn a B on the STAT 250 final exam. Under Applets, open the Sampling
distribution applet (box shown below). First, select Binary for the population, then enter the
value for p = 0.175, the proportion of students who earn a B in the box next to “p:” Then click on
Compute.
a) Once the applet box is opened, enter 10 in the box to the right of the words “sample size”
in the right middle of the applet box window (see image below). Then, at the top of the
applet, click “1 time.” Watch the resulting animation. When the sample is completed,
copy and paste the entire applet box (using options copy) into your document.
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b) Click Reset at the top of the applet. Retype 10 in the sample size box. Then, click the
“1000 times” to take 1000 samples of size 10. Copy and paste the applet image into your
document. Describe the shape of the Sample Proportions graph at the bottom of your
image in one sentence. Why do you think that this graph does not have a bell shape?
Use the central limit theorem conditions to answer this question in another sentence.
c) Click Reset at the top of the applet. Type 100 in the sample size box. Then, click the
“1000 times” to take 1000 samples of size 100. Copy and paste the applet image into
your document. Describe the shape of the Sample Proportions graph at the bottom of
your image in one sentence. Why do you think that this graph has the shape you
described? Answer this question in another sentence.
d) Using the image in part (c), record the values you obtained for the mean (in green) and
the standard deviation (in blue). What does each of these values estimate in the sampling
distribution? Answer this question in one sentence.
e) Now calculate the standard error using p = 0.175 and n = 100. Compare this value to the
standard deviation you recorded in part (d) in one sentence.
Problem 4: Confidence Interval for Percentage of Bs.
The data set “STAT 250 Final Exam Scores” contains a random sample of 385 STAT 250
students’ final exam scores (maximum of 100) collected over the past two years. Answer the
following questions using this data set.
a) Calculate the sample proportion from our sample of 385 Final Exam Scores to determine
the proportion of students who earned Bs on the final exam. Consider a B between 80
and 89. Use StatCrunch to obtain this value and display the value in your document.
There are multiple approaches to obtaining this value. One possible way would be to sort
the “Scores” column and count the number of scores between 80 and 89.
b) Using the sample proportion obtained in (a), construct a 95% confidence interval to
estimate the population proportion of students who earned a B on the final exam. Please
do this “by hand” using the formula and showing your work (please type your work, no
images accepted here). Then, verify your result using Stat Proportions Stats One
Sample With Summary. Copy and paste your StatCrunch result in your document as
well.
c) Interpret this confidence interval in one sentence using the structure learned in class.
d) Did this confidence interval capture the population proportion given in Problem 3 (p =
0.175). Answer this question in one sentence.
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e) Use the confidence interval applet in StatCrunch to simulate constructing 1000 95%
confidence intervals using p = 0.175 and n = 385. Once the window is open, click reset
and click 1000 intervals. Copy and paste your image into your document.
Box 1: Enter the given
population proportion, 0.175
Box 2: Enter the given
confidence level 0.95
Box 3: Enter the given sample
size, n=385
f) Using this image and specifically the “Proportion (Prop.) contained” value, write a longrun interpretation for your confidence interval method in one sentence.
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