All The answers need to be as instructed in the document below. They all need to be typed in and if it requires graph, it needs to be obtained through statcrunch.

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STAT 250 Spring 2019 Data Analysis Assignment 3

1.

Number your pages across your entire solutions document.

2.

Your document should include the ANSWERS ONLY with each answer labeled by its

corresponding number and subpart. Keep the answers in order. Do not include the

questions in your submitted document.

3.

Generate all requested graphs and tables using StatCrunch.

Elements of good technical writing:

Use complete and coherent sentences to answer the questions.

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.

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 are not accepted on this assignment.

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.

Problem 1: Confidence Interval for Percentage of A’s.

The data set “STAT 250 Final Exam Scores” contains a random sample of 269 STAT 250

students’ final exam scores (maximum of 80) collected over the past two years. Answer the

following questions using this data set.

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Here is the data:

Scores

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a) What proportion of students in our sample earned A’s on the final exam? A letter grade

of A is obtained with a score of 72 or higher. Hint: You can do this many ways, but in

StatCrunch, go to Data → Row Selection → Interactive Tools. In the slider selectors

box, click the variable “Scores” into the variable box. Then click compute. Use the

slider to obtain the count by looking at the “# rows selected” presented in the first line of

the box. Show your work (i.e. describe the method you used to obtain the number of A’s)

and express this value as a proportion rounded to four decimal places.

b) Assuming there is a big population of students who have completed the final exam in

STAT 250, write one sentence each to check the two other conditions of the Central

Limit Theorem.

c) Using the sample proportion obtained in (a), construct a 95% confidence interval to

estimate the population proportion of students who earned an A on the final exam. Please

do this “by hand” using the formula and showing your work (please type your work, no

images accepted here). Round your confidence limits to four decimal places.

d) Verify your result from part (b) using Stat → Proportions Stats → One Sample → With

Summary. Inside the box, enter the number of students who earned an A as the # of

successes, the sample size as the # of observations, and select confidence interval and

click Compute! Copy and paste your StatCrunch result in your document.

e) Interpret the StatCrunch confidence interval in part (c) in one sentence using the context

of the question.

f) Use the Confidence Interval applet (for a Proportion) in StatCrunch to simulate

constructing one thousand 95% confidence intervals assuming the proportion of A’s in

the population is p = 0.11 and the sample size n = 269. Once the window is open, click

reset and select (or click) 1000 intervals. Copy and paste your image into your

document.

7

Box 1: Enter the given

population proportion, 0.11

Box 2: Enter the given

confidence level 0.95

Box 3: Enter the given sample

size, n=269

g) Compare the “Prop. contained” value from part (f) to the confidence level associated with

the simulation in one sentence.

h) Write a long-run interpretation for your confidence interval method in context in one

sentence.

Problem 2: Food Delivery Robots

As we all know, GMU began a robot food delivery service in January. One of the potential

benefits of this service is to help the busiest students eat breakfast. Research has shown that

about 88% of college students skip breakfast due to busy schedules and other reasons. Initial

data were collected from a random sample of 291 Mason students who utilize the robot food

delivery service and are presented in StatCrunch. The responses (0 = ate breakfast and 1 = did

not eat breakfast) are found in StatCrunch in a data set called “Food Delivery Robots.”

Here’s The data:

Breakfast

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a) Obtain the sample proportion of individuals who said “they did not eat breakfast” using

Stat → Tables → Frequency in StatCrunch. Only the value of the sample proportion is

needed in your answer. Present this sample proportion as a decimal rounded to 4 decimal

places.

b) Using = 0.05, is there sufficient evidence to conclude that less than 88% GMU students

who utilize the food delivery robots skip breakfast? Conduct a full hypothesis test by

following the steps below.

14

i.

ii.

iii.

iv.

v.

vi.

vii.

viii.

ix.

Define the population parameter in one sentence.

State the null and alternative hypotheses using correct notation.

State the significance level for this problem.

Check the three conditions of the Central Limit Theorem that allow you to use

the one-proportion z-test using one complete sentence for each condition. Show

work for the numerical calculation. Assume the population is large.

Calculate the test statistic “by-hand.” Show the work necessary to obtain the

value by typing your work and provide the resulting test statistic. Do not round

while doing the calculation. Then, round the test statistic to two decimal places

after you complete the calculation.

Calculate the p-value using the standard Normal table and provide the answer.

Use four decimal places for the p-value.

State whether you reject or do not reject the null hypothesis and the reason for

your decision in one sentence (compare your p-value to the significance level to

do this).

State your conclusion in context of the problem (i.e. interpret your results and/or

answer the question being posed) in one or two complete sentences.

Use StatCrunch (Stat → Proportion Stats → One Sample → with Data) to verify

your test statistic and p-value. Copy and paste this box into your document.

Problem 3: GMU Shuttle Service

GMU officials are trying to determine if they are using the correct number of campus shuttles for

the number of individuals who use them. If the proportion of individuals associated with GMU

(including faculty, staff, and students) who use the shuttles is significantly different from 0.28,

the officials believe they will have to either remove or add a shuttle to the fleet. In a random

sample of 444 people taken from the population of all individuals associated with GMU

(including faculty, staff, and students) it was discovered that 123 of these individuals use the

shuttle.

a) Check the three conditions of the Central Limit Theorem that allow you to use the oneproportion confidence interval using one complete sentence for each condition. Show

work for the numerical calculation.

b) Construct a 99% confidence interval to estimate the population proportion of these

individuals who use the shuttle system. Calculate this “by hand” using the formula and

showing your work (please type your work, no images accepted here). Round your

confidence limits to four decimals.

c) Verify your result in part (b) using Stat → Proportions Stats → One Sample → With

Summary. Copy and paste your StatCrunch result in your document as well.

d) Using = 0.01, is there sufficient evidence to conclude that the proportion of the

individuals who use GMU shuttles is different from 0.28? Conduct a full hypothesis test

by following the steps below. Enter an answer for each of these steps in your document.

15

i.

ii.

iii.

iv.

v.

vi.

vii.

viii.

Define the population parameter in one sentence.

State the null and alternative hypotheses using correct notation.

State the significance level for this problem.

Calculate the test statistic “by-hand.” Show the work necessary to obtain the

value by typing your work and provide the resulting test statistic. Do not round

during the calculation. Then, round the test statistic to two decimal places after

you complete the calculation.

Calculate the p-value using the standard Normal table and provide the answer.

Use four decimal places for the p-value.

State whether you reject or do not reject the null hypothesis and the reason for

your decision in one sentence (compare your p-value to the significance level to

do this).

State your conclusion in context of the problem (i.e. interpret your results and/or

answer the question being posed) in one or two complete sentences.

Use StatCrunch (Stat → Proportion Stats → One Sample → with Summary) to

verify your test statistic and p-value. Copy and paste this box into your

document.

e) Explain the connection between the confidence interval and the hypothesis test in this

problem (discuss this in relation to the decision made from your hypothesis test and

connect it to the confidence interval you constructed in part (b)). Answer this question in

one to two sentences.

Problem 4: Building another Sampling Distribution

We will use the Sampling Distribution applet in StatCrunch to investigate properties of sampling

distributions of the mean for a right skewed distribution. Under Applets, open the Sampling

distribution applet (box shown below). First, select “right skewed” for the population and then

click on Compute.

a) Once the applet box is opened, enter 5 in the box to the right of the words “sample size”

in the right middle of the applet box window. 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.

b) Click Reset at the top of the applet. Then, click the “1000 times” to take 1000 samples of

size 5. Copy and paste the applet image into your document.

c) Describe the shape of the Sample means graph at the bottom of your image from part (b)

in one sentence.

d) Why do you think that this graph does not have an approximately Normal shape? Use the

Central Limit Theorem large sample size condition (for means) to answer this question in

one sentence.

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e) Click Reset at the top of the applet. Type 40 in the sample size box. Then, click the

“1000 times” to take 1000 samples of size 40. Copy and paste the applet image into your

document.

f) Describe the shape of the Sample means graph at the bottom of your image from part (e)

in one sentence.

g) Why do you think that this graph from part (f) has the shape you described? Use the

Central Limit Theorem large sample size condition to answer this question in one

sentence.

h) Using the image in part (e), write the values you obtained for the mean (in green) and the

standard deviation (in blue). These values are found in the bottom right box labeled

“Sample Means”

i) Compare the mean value (in green, found in part (h)) to the known mean of the

population from the top box labeled “Population.”

j) Now calculate the standard error of the sample mean using the value labeled “Std. dev.”

in blue from the top box. Round this value to three decimal places.

k) Compare the value in part (j) to the standard deviation (in blue) you obtained in part (h)

in one sentence.

l) Assuming this right skewed population distribution had a population mean of 14.05 and a

standard deviation of 11.83; calculate the probability that, in a random sample of 30, the

mean of the sample is greater than 16. First, draw a picture with the mean labeled, shade

the 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.

m) Verify your answer in part (l) using the StatCrunch Normal calculator and copy that

image into your document. In addition, write one sentence to explain what the

probability means in context of the question.

17

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