Activity 4
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School
Oregon State University, Corvallis *
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Course
202
Subject
Mathematics
Date
Apr 3, 2024
Type
Pages
3
Uploaded by CommodoreCat4790
ST202 Recitation Activity 4:
z
procedures for proportions
40 Points
Instructions
Instructions:
●
Clearly label and
type answers
to the questions on the proceeding pages in Word, Google Docs, or other word
processing software.
●
Insert
diagrams or plots as a picture
in an appropriate location.
●
Math Formulas need to be typed with Math Type, LaTeX, or clearly using key board symbols such as +, -. *, /, sqrt()
and ^
●
Submit assignment to the Gradescope link as a PDF. Indicate the pages to the individual questions and also verify the
correct document has been uploaded. Failing to follow this direction may result in point deductions.
o
If you are unable to answer any questions before the submission deadline, please still match the question
location to the correct page when submitting in Gradescope.
Inference for a single proportion.
A university claims that 50% of its students have paid summer internships the summer between their freshman
and sophomore year. You are skeptical and think the percentage of students that have a paid summer internship
between their freshman and sophomore year is less than 50%. To test this claim, you randomly select 20
students from the university and survey them to find out if they had a paid internship between their freshman
and sophomore year. You find that only 8 of the 20 students had a paid internship.
Perform one-sample
z
test to test the question of interest. Use a significance level of 0.05 Follow the four steps
of a complete hypothesis test and include a four-part conclusion.
1. State:
(4 point) What is the question of interest?
What is the estimated population proportion of students who had a paid summer internship between their freshman and
sophomore year?
2. Plan:
(4 point) What is the parameter of interest?
The students who had a paid internship between their freshman and sophomore year.
3.
(4 points) State the null and alternative hypotheses.
H
0
: p = 0.50
H
a
: p < 0.50
4.
(1 point) State the significance level.
α = 0.05
5. Solve:
(3 point) Check the conditions.
Random?
The sample is representative of the population of interest as the surveyed students were randomly selected.
np
0
> 10 and n(1 - p
0
) > 10?
As this is only used for large sample sizes, it does not need to be used as there were only 20
people surveyed.
**Regardless of whether the conditions were met or not, continue with the problem.**
6.
(4 points) Calculate the test statistic.
Show your work.
p-hat = 8/20 = 0.4
z = (p-hat - p
0
) / √(p
0
(1 - p
0
) / n)
z = (0.4 - 0.5) / √(0.5 (1 - 0.5) / 20)
z = -0.8944
7.
(4 points) Find the p-value.
p-value = 0.1857
8.
(6 points) Calculate and interpret a 95% confidence interval for the true proportion of students that had a paid internship
between their freshman and sophomore years at this university.
Show your work.
p-hat +/- z* √(p-hat (1 - p-hat) / n)
Lower Bound = p-hat - z* √(p-hat (1 - p-hat) / n) = 0.4 - 1.96 √(0.4 (1 - 0.4) / 20) = 0.1853
Higher Bound = p-hat + z* √(p-hat (1 - p-hat) / n) = 0.4 - 1.96 √(0.4 (1 - 0.4) / 20) = 0.6147
(0.1853, 0.6147)
The 95% confidence interval for the estimated proportion of students who had a paid summer internship between their
freshman and sophomore year is from 0.1853 to 0.6147.
9. Conclude:
(8 points) Use the four-part conclusion method.
There is no evidence that the true population proportion of students who
had a paid summer internship between
their freshman and sophomore year is less than 50%. The null hypothesis from a one-sided z-test for proportions failed to
be rejected with a p-value of 0.1857.
The estimated proportion of students who had a paid summer internship between their freshman and sophomore year is
0.4 with a 95% confidence interval of 0.1853 to 0.6147. There is no evidence that the
students who
had a paid summer
internship between their freshman and sophomore year is less than 50%
10. Gradescope Page Matching (2 points)
When you upload your PDF file to Gradescope, you will need to match each question on this assignment to the
correct pages. Video instructions for doing this are available in the Start Here module on Canvas on the page
“Submitting Assignments in Gradescope”. Failure to follow these instructions will result in a 2-point deduction
on your assignment grade. Match this page to outline item “Gradescope Page Matching”.
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