QUANT REASON WITH STATS

 

Links to Applets we will be using: 

• http://www.rossmanchance.com/applets/OneProp/OneProp.htm?candy=1 (Links to an external site.) 
• https://sample-size.net/confidence-interval-proportion/ (Links to an external site.) 
• https://digitalfirst.bfwpub.com/stats_applet/generic_stats_applet_22_pvalprop.html (Links to an external site.) 

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TAMS #4 Assignment:
Confidence Intervals and Hypothesis testing

Purpose: In this assignment, you will explore how we can make inferences about unknown population
parameters, the meaning of the margin of error, and how predictions in the press about a result with a
margin of error are to be interpreted. In the second problem, you will perform a hypothesis test to
decide whether the claim made is supported by the data or is too unlikely to happen under the null
hypothesis.

The mastery standards covered by this assignment are

S9 – Confidence Intervals: Students will be able to compute confidence intervals and interpret the
interval in the context of the problem.

S10 – Hypothesis Testing: Students will be able to perform hypothesis tests for proportions or means
and interpret the final result in the context of the problem.

MP2 – Communicate a Viable Argument: Students will be able to justify a statistical analysis in words
and communicate that justification.

Instructions:

Note that this is an INDIVIDUAL assignment, and you are responsible for doing your own write-up,
even though you may have started to work on this TAMS in class as a pair or group. You are allowed to
discuss this assignment with other class members, but you may NOT work together on an electronic
document. “Working together” in this context is ONLY by talking, not by typing. You MUST do your
own simulations and produce your OWN written solution. Documents that begin as a shared
document and are modified will NOT be accepted and will result in a mastery grade of 0 for everybody
involved.

DO NOT SUBMIT ANY ANSWERS POSTED ON CHEGG. ANSWERS THAT MATCH CHEGG WILL BE
REPORTED AS A VIOLATION OF THE ACADEMIC HONESTLY POLICY. DO NOT POST ANY SCREENSHOTS
OR DOCUMENTS RELATED TO THIS ASSIGNMENT ON CHECK. POSTING IS A VIOLATION OF BOTH THE
ACADEMIC HONESTY POLICY AND THE INTELLECTUAL PROPERTY RIGHTS OF THE AUTHOR.

You are to submit your answers online through Canvas using the template posted with the assignment.

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Problem 1 – Exploring Confidence intervals

Confidence intervals are at the heart of statistical inference. They tell us how confident we can be that
the data we get from a sample truly represents what the population parameter is. Whether it is
predicting the outcome of an election, doing market analysis to project the popularity of a consumer
product, or testing a medical device for safety and efficacy, confidence intervals allow us to have a
sense of what the world looks like.

You are a manager at The Hershey Company in charge of manufacturing bags of Reese’s. Each bag has
a total of 200 pieces of candy in it.

a) Your boss tells you not to have fewer than 20% and not more than 45% of the Reese’s Pieces be

orange, but within that range, you get to select. Make a selection of p, the desired manufacturing
proportion of orange Reese’s Pieces from the interval [0.20, 0.45], using two decimal places. Write
down this value of p at the top of the answer template.

b) Once production of the bags of Reese’s Pieces has started, quality control needs to test that the
manufacturing proportion is happening as designed. We’re going to use a Reese’s Pieces app to
draw random samples of mixed bags of candy. Go to
http://www.rossmanchance.com/applets/OneProp/OneProp.htm?candy=1 and set the probability
of orange pieces to your choice of the desired manufacturing proportion p (with two decimal
places), the number of candies to 200 and the number of samples to 1. Uncheck the box next to
“Animate”. Then click on Draw Samples. Write down the number of orange pieces in the second
column of the table in the answer template. Then compute the 95% confidence interval and fill in
the information in the first row of the table of the answer template. Remember – z* for 95%
confidence level is 1.96.

• # of orange candies in your sample: _______ (record in column 1 of the table)

• Sample proportion: �̂� = _______ (record in column 2 of the table)

• Standard error: #!”($%!”)
‘

=

• Margin of error: ± 𝑧∗ ∙ #!”($%!”)
‘

=

• Lower bound of 95% confidence interval: �̂� − 𝑧∗ ∙ #!”($%!”)
‘

=

• Upper bound of 95% confidence interval: �̂� + 𝑧∗ ∙ #!”($%!”)
‘

=

• Interval in which the true parameter lies with 95% probability:

Either take a photo of your computations and insert it into the answer template or type your
calculations into the answer template using the Word equation editor.

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c) Now go back to the App and create 19 bags of 200 candies, one at a time. After each time you get a
bag (sample), record the number of orange pieces in the first column of the table in the answer
template.

d) You will now use an app for confidence intervals https://sample-size.net/confidence-interval-
proportion/ which will perform the computations you did in part b by hand for the 19 bags you
created in part c. Follow the example in the app instructions document to set up the app and to
compute the sample proportions and confidence intervals for the 20 bags. Read off the sample
proportions, record them, and use the lower and upper limits of the confidence intervals to write
the confidence interval in the table in the format (lower limit, upper limit) in the third column of
the table. Round to three decimal places.

Finally, for each row, check whether the value of the desired manufacturing proportion p that you
selected (listed at the top of the answer template) is contained in the confidence interval or not. If
it is in the interval, put a checkmark into the last column. Otherwise, put an x. Make sure you are
checking for p and not p-hat.

e) Count how many of the confidence intervals in your table contain your chosen parameter value p
(the desired manufacturing proportion), the true proportion of orange pieces that is included in the
bag. Make sure to count each interval separately. In other words, if you had two different bags that
had the same number of orange pieces, you count those intervals separately. Once you have counted
the intervals that contain p, compute the percentage of intervals (of the 20) that contain your chosen
value of p. Show your computation. Give your final percentage in the context of the problem.

f) How does your answer in part e relate to the fact that you computed confidence intervals at level

95%? Write a short paragraph explaining what it means to have a 95% confidence interval in terms
of the proportion of orange pieces that you can expect in any given bag of candy.

Problem 2 – Hypothesis Testing

California has recently been a battleground regarding whether or not “gig economy workers”,
particularly drivers for companies like Uber and Lyft, should be classified as employees or independent
contractors. Being classified as an employee has benefits such as overtime pay and worker’s
compensation insurance. Being classified as an independent contractor has benefits such as flexibility
and the ability to write off expenses on your taxes. Uber and Lyft say that most of their drivers need
the flexibility, as they drive part-time to supplement a second job.

A worker’s advocacy group is concerned that despite the companies’ comments that drivers are
actually driving for Uber and Lyft as their primary job. The group defines the job as a primary job if
income from Uber and Lyft accounts for more than half of a worker’s income. The advocacy group
decides to commission a survey of Uber and Lyft drivers to determine if more than 60% of the drivers
are driving as their primary job or as a secondary job.

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The advocacy group did a study of 1,127 Uber drivers and found that 714 of them were driving as their
primary job. They then did a study of 823 Lyft drivers and found that 510 of them were driving as their
primary job.

Conduct an appropriate hypothesis test for each company to check whether the results from this
sample provide sufficient evidence that most of the drivers for Uber and/or Lyft are driving as their
primary job.

a) What is the most appropriate test, a two-tailed test, a right-tailed test, or a left-tailed
test? Explain how you know.

b) State the null and alternative hypotheses. Make sure to use proper notation.

c) Use an app or calculator/z-score table to do the following:

i. Calculate the test statistic.

ii. Determine the P-value

An app option is:
https://digitalfirst.bfwpub.com/stats_applet/generic_stats_applet_22_pvalprop.html
If you use an app, Insert a screenshot for each hypothesis test from the app into your answer
template and explain how the app answers your question.

d) State the conclusion about the null hypothesis.

e) For each company, do more than 60% of their drivers drive as a primary job? Justify your

answer with evidence from the hypothesis test.

S21 TAMS 4 Answer template Name: _____________________________

#1

a)

p = ____________

b)

c)

&

d)

# of orange pieces	Sample proportion	Confidence Interval	Is p in the confidence interval? (ü or x)

e)

f)

#2

Uber

a)
b)
c)

Include screenshot from the app here:

d)
e)

Lyft

a)
b)
c)

Include screenshot from the app here:

d)
e)