A list of 241 public colleges and universities in the States of Georgia (GA), South Carolina (SC),
North Carolina (NC), Tennessee (TN), Kentucky (KY), Virginia (VA), West Virginia (WV),
Maryland (MD), Delaware (DE), Pennsylvania (PA), New Jersey (NJ), New York (NY),
Connecticut (CT), Rhode Island (RI), and Massachusetts (MA) and their In-State Tuition was
collected. The data set found in our StatCrunch group is called “In-State Tuition” and the
variables State, University, and In-State Tuition (in dollars) are presented. Consider the 241
observations as a sample of all public colleges and universities in the United States.
a) Use StatCrunch to construct an appropriately titled and labeled relative frequency
histogram of the “In-State Tuition” variable. Copy your histogram into your solutions.
b) What is the shape of this distribution? Answer this question in one complete sentence.
c) Now overlay your highlighted histogram from part (a) with a Normal curve and add a
vertical line at the mean. Go 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 solutions.
d) Do you think it is reasonable to use the Normal probability model in this case? Answer
this question and provide a reason why in one sentence.
e) Using your histogram, determine the proportion of Universities with In-State Tuition less
than $5,000. To do this, click on each bar that is representing less than $5,000 and look
in the bottom left of your screen to see how many Universities are highlighted. Calculate
the correct proportion and type your work for the calculation of the proportion. Please
round your proportion to four decimal places.
f) Calculate the mean and the standard deviation of the “In-State Tuition” variable using
StatCrunch. (Select Stat Summary Stats Columns.) Copy and paste a table only
presenting the mean and standard deviation into your solutions. Round the mean and
standard deviation to two decimal places inside this table.
No matter your answer to part (d), for parts (g) – (k), assume that the distribution of In-State
Tuition in the population is Normal with the mean and standard deviation found in part (f). (Use
the rounded mean and standard deviation values.)
g) Use the Standard Normal Table (Table 2 in your text or formula packet) to calculate the
probability that a randomly selected College will have an In-State Tuition of less than
$5,000. Type all calculations needed to find this probability and your answer in your
solutions.
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h) Verify your answer in part (g) using the StatCrunch Normal calculator by using: Stat
Calculators Normal. Use the mean and standard deviation found in part (f) and copy
the image with all values included into your solutions. Make sure your image presents all
values including the probability. In addition, write one sentence to explain what the
probability means in context of the question posed in part (g).
i) Compare the probability from part (h) to the proportion you calculated in part (e) using
the context of the question. Label each value as a theoretical or an empirical probability
in your comparison.
j) Use StatCrunch Normal calculator to calculate the probability that that a randomly
selected University will have an In-State Tuition between $10,000 and $15,000. Present
your StatCrunch image as your answer as you did in part (h).
k) Calculate the minimum tuition that would put a public university in the highest 3% (i.e.
top 3%) of all in-state tuitions. Provide the minimum tuition in dollars and cents such
that this value of any dollar amount higher would be in the top 3% of all tuitions. First,
type all calculations using the Standard Normal Table necessary to obtain your answer.
Round your answer to two decimal places (i.e. dollars and cents). Then, verify your
result using the StatCrunch Normal calculator and copy the StatCrunch image in your
solutions as you did in part (h).
Investigation 2: Bachelor’s Degrees around the DMV (no data set)
From the U.S. Census Bureau’s American Community Survey in 2019 it was found that 33.13%
of United States residents over the age of 25 had an educational attainment of a bachelor’s
degree or higher. In the District of Columbia, the percentage of residents over the age of 25 who
had attained a bachelor’s degree or higher was 59.67%. An investigator for the U.S. Census
Bureau took a random sample of seven residents from the District of Columbia and asked them
their highest educational degree they had obtained. When there is no data set posted for an
investigation, open a blank StatCrunch page by clicking “Open StatCrunch” on the StatCrunch
home page.
a) Verify that the sample from the District of Columbia satisfies the conditions of the
binomial experiment. Write one sentence to check each requirement in context of the
investigation.
b) Assuming the sample from the District of Columbia is a binomial experiment, build the
probability distribution in a single table and include the table in your solutions. You may
present this table horizontally or vertically and leave the probabilities unrounded. There
are two possible ways to do this:
- You may use Data Compute Expression and choose the function dbinom. This
method relies on you entering all the outcome values of the random variable in the
first column of your data table. Copy the table from StatCrunch to your answer
solutions.
4 - 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 = 7. Then create the
table in your solutions based on these results.
c) Calculate the probability that at least five individuals from the District of Columbia in
this sample have attained a bachelor’s degree or higher using the probability distribution
table you created in part (b). Type all of your calculations and use proper probability
notation in your solutions. Round your final answer to four decimal places.
d) Verify your answer to part (c) using the StatCrunch binomial calculator. Copy the full
image into your solutions. Write a one-sentence interpretation of the probability in
context of the question.
e) Assume we take a sample of seven United States residents and this sample has a binomial
distribution. Calculate the probability that at least five residents from the United States in
this sample have attained a bachelor’s degree or higher using the StatCrunch binomial
calculator. Copy the image of the Binomial calculator in your solutions. In addition,
write a one-sentence interpretation of the shape of the probability distribution graph.
f) Compare the probability calculated in part (e) to the probability you calculated in part (d)
in one sentence.
g) Calculate the mean and standard deviation of the number of individuals from the District
of Columbia in this sample who have obtained a bachelor’s degree or higher. Show your
work using the binomial mean and binomial standard deviation formulas in your
solutions. Round your answers to two decimal places. (It is not necessary to use
StatCrunch for this part.)
h) Imagine you repeated taking a sample of seven individuals from the District of Columbia
population 100 times. We can simulate this in StatCrunch. First, go to Data Simulate
Binomial. Next, enter 100 for Rows, 1 for Columns, 7 for n, 0.5967 for p, and click
compute. Then, to visualize these data, go to Graph Histogram. Produce a properly
titled and labeled frequency histogram and paste it into your solutions.
i) Calculate the proportion of the 100 samples that had at least five District of Columbia
residents having obtained a bachelor’s degree or higher. Round your answer to four
decimal places and show your calculation in your solutions.
j) In one sentence, compare the proportion of the 100 samples having at least five residents
with a bachelor’s degree or higher (calculated in part (i)) with the probability of having at
least five residents with a bachelor’s degree or higher (calculated in parts (c) and (d)). In
your comparison, define each probability as empirical or theoretical.