Math 10 - Lab 3 - Spring 2023
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School
De Anza College *
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Course
10
Subject
Mathematics
Date
Feb 20, 2024
Type
Pages
7
Uploaded by SuperHumanCrown9797
Math 10
Bambhania
Spring 2023
Lab 3: Inferential Statistics
DIRECTIONS: Fill in your answers in
bold
or in
color
.
The data in the shared spreadsheet is the same data that you used for Lab 1. It was collected
from my 2 different Math 10 classes at De Anza this quarter (Spring 2023): one asynchronous
online class and one in-person class with co-requisite.
You were sent an email with the subject ‘Data File for Lab 3’. We will use the data file, titled
Student Data Collection – Spring 2023.xlsx, in this lab.
IMPORTANT
: Before starting the lab, you need to open this file from your email in your Google
Drive as a Google Sheet.
If you simply preview it from the email, you won’t be able to run the
commands needed for the lab.
PART I: CONFIDENCE INTERVALS
Confidence Intervals online calculators:
Although you have constructed confidence intervals by hand thus far, using formulas, you may
use the following calculators in this lab as a shortcut.
●
Confidence Intervals for Proportions:
https://www.statskingdom.com/proportion-confidence-interval-calculator.html
●
Confidence Intervals for Means:
https://www.statskingdom.com/confidence-interval-calculator.html
1.
In the survey, students were asked “In which modality of class do you learn the best?”.
Let’s examine the results of that question. The data for this question is in Column C of
the worksheet.
a.
Construct a 90% confidence interval estimating the proportion of all Spring 2023
De Anza College Math 10 students who learn better in a modality with “Some
in-person component”. Follow these steps:
●
In the spreadsheet, compute x, the “number of successes” by typing the
command
’=countif(
C2:47
,"Some in-person component")’
at the
bottom of Column C. Note that the sample size is
46
.
●
Again, instead of constructing the confidence interval by hand, go to the
appropriate confidence interval calculator (links above this problem), and
find the confidence interval. Use the result using the ‘Normal
approximation’ method.
●
Paste your result below.
p̂ = 0.4783.
Confidence interval: [0.3571 , 0.5994].
Alternatively: 0.4783 ± 0.1211
The margin of error (MOE) or estimated bound on proportion (EBP) is
0.1211 or 12.1146%
b.
Interpret your confidence interval in the context of the problem.
We estimate with 90% confidence that the true proportion of students who learn
better in a modality with "Some in-person component" falls between 35.71% and
59.94%.
c.
Use your confidence interval to find the point estimate (sample proportion) and
the margin of error for this confidence interval.
= 0.4783, which suggests that approximately 47.83% of Math 10
0.3571+0.5994
2
students in Spring 2023 at De Anza College learn better in a modality with "Some
in-person component."
d.
If we were to increase the confidence level to 95%, what would happen to the
margin of error? Increase, decrease, or stay the same? Explain.
The margin of error would increase. It would result in a wider confidence interval
and a larger margin of error. The wider interval is needed to capture a larger
range of possible values for the true proportion. This increase in uncertainty is
due to the higher confidence level requiring a more conservative approach.
Therefore, the margin of error would increase as we aim to provide a more
confident estimation.
2.
In the survey, students were asked “On average, how much does it cost you to get a
haircut (including tip)?”. Let’s examine the results of that question. This information is
stored in Column P.
a.
We wish to construct a 99% confidence interval estimating the mean amount of
money that Spring 2023 De Anza Math 10 students spend on a haircut (including
tip) using our sample.. Should we use the normal distribution or the Student-t
distribution? Explain why.
We should use a student-t distribution since we are using a sample to estimate
the mean, it is unlikely that we know the population standard deviation. The
Student-t distribution is appropriate when the population standard deviation is
unknown and must be estimated from the sample.
b.
Construct the 99% confidence interval. Follow these steps:
●
Go to the appropriate confidence interval calculator (links above Question
1). Again, instead of constructing the confidence intervals by hand, we will
use a calculator shortcut.
●
In the calculator, pull down the menu for ‘Data is’ and select ‘Raw Data’ to
enter data. Copy the data from Column P (just the numbers - that is, copy
cells P2 through P47) and paste them.
●
Fill in the other entries appropriately and hit ‘Calculate’.
●
Paste your results below.
Mean confidence interval: [19.6434 , 43.4262].
Alternatively: 31.5348 ± 11.8914
Margin of Error (MOE): 11.8914.
Standard Error (S.E): 4.4213.
c.
Interpret your confidence interval in the context of the problem.
We are 99% confident that the true population mean lies within 19.64% and
43.43%.
d.
If we were to gather data from more Spring 2023 De Anza Math 10 students to
increase our sample size, what can we expect to happen to the margin of error:
Increase, decrease, or stay the same? Explain.
Margin of arrow will decrease because the margin of error is influenced by the
sample size and the variability of the data. As the sample size increases, the
estimate becomes more precise, and the variability in the data has a smaller
impact on the margin of error.
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PART II: HYPOTHESIS TESTS
Hypothesis Tests for Single Mean and Proportion online calculators:
●
One Sample Z-Test:
https://www.statskingdom.com/110MeanNormal1.html
●
One Sample T-Test:
https://www.statskingdom.com/130MeanT1.html
●
One Sample Proportion Test: https://www.statskingdom.com/111proportion_normal1.html
3.
In the Spring 2023 survey, students were asked “On average, how many hours of sleep
do you get per night?”. Let’s analyze how Spring 2023 De Anza Math 10 students’
answer compares to that for all American college students. Nationally, the mean number
of hours of sleep that college students get is 6.5. Is the mean number of hours of sleep
that Spring 2023 De Anza Math 10 students get different from 6.5? Test at 5%
significance level.
a.
State and define the appropriate population parameter that this test is about.
= the population mean number of hours of sleep for Spring 2023 De Anza Math
µ
10 students
b.
State the appropriate null and alternative hypotheses.
H
0
:
H
a
:
µ = 6. 5
µ ≠ 6. 5
c.
Should we use the Normal distribution or a Student t-distribution to run this test?
Why?
We should use the student t-distribution since we don’t know the population
standard deviation
d.
Run the hypothesis test at the correct online calculator (links above Question 3).
Follow these steps:
●
Put in the correct H
a
and the significance level.
●
Put ‘Expected mean’ based on your H
0
. Leave other items as they are.
●
Select on ‘Enter raw data directly’, and copy the data from Column L
(again, just the numbers) and paste them. Then use ‘Calculate Sample’
button.
●
Click ‘Calculate.
Paste the results below. Be sure to include the sample mean, sample standard
deviation, test statistic and the p-value.
P-value = 0.2284
t = 1.2212
Sample size (n) = 46
Average (x̄ ) = 6.7391
Standard deviation (S) = 1.3281
e.
Sketch an accurate and well-labeled graph on a piece of paper that shows the
p-value - similar to what we draw in this class. Paste it here.
f.
What is your decision for this hypothesis test? Write the conclusion for this
hypothesis test in the context of the problem.
We fail to reject the null
We conclude that we do not have enough evidence to support the claim that the
mean number of hours of sleep for Spring 2023 De Anza Math 10 students is
different from the mean number of hours of sleep for all American college
students.
Hypothesis Tests for Comparing Two Means and Two Proportion online calculators:
●
Paired T-Test:
https://www.statskingdom.com/paired-t-test-calculator.html
●
Two Sample Z-Test:
https://www.statskingdom.com/120MeanNormal2.html
●
Two Sample T-Test:
○
Pooled:
https://www.statskingdom.com/140MeanT2eq.html
○
Not Pooled: https://www.statskingdom.com/150MeanT2uneq.html
●
Two Sample Proportions Test:
https://www.statskingdom.com/121proportion_normal2.html
4.
In the Spring 2023 survey, students were asked:
●
On average, how many HOURS do you spend towards your studies PER
WEEK?
●
On average how many HOURS do you work (paid employment) PER WEEK?
Let’s use a hypothesis test to answer the question: Given the high cost of living in the
Bay Area, do Spring 2023 De Anza College Statistics students work MORE THAN they
study? Use a significance level of 5%.
a.
Describe the appropriate parameter(s) of interest for this test.
HINT: Think
carefully
about whether this is a case of independent sampling or
dependent sampling.
This is a dependent sampling because each student's study hours and work
hours are paired or linked to the same individual. (
= work
= study)
µ
𝑤
µ
𝑠
= The population mean of the number of hours spent towards studies per
µ
𝑠
week for all Spring 2023 De Anza College Statistics students.
= The population mean of the number of hours spent working (paid
µ
𝑤
employment) per week for all Spring 2023 De Anza College Statistics students.
b.
State the appropriate null and alternative hypotheses.
H
0
:
H
a
:
µ
𝑤
≤ µ
𝑠
µ
𝑤
> µ
𝑠
c.
The necessary sample data are in column H and column I of the spreadsheet.
Run the hypothesis test at the correct online calculator (links above Question 4).
Paste the output - be sure to include the test statistic and the p-value.
P-value = 0.1482
t = 1.0565
Sample size (n) = 46
Average of differences (x̄ d) = 3.7826
SD of differences (Sd) = 24.2825
d.
Sketch an accurate and well-labeled graph on a piece of paper that shows the
p-value. Paste it here.
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e.
What is your decision for this hypothesis test? Write the conclusion for this
hypothesis test in the context of the problem.
Fail to reject the null hypothesis.
There is not enough evidence to conclude that Spring 2023 De Anza College
Statistics students work more than they study, given the high cost of living in the
Bay Area.