10.1.5: Hypothesis Testing
- Page ID
- 63626
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Semester Project – Assignment 5: Hypothesis Testing – Comparing Lakewood and Golden
In this assignment, you’ll conduct a formal hypothesis test comparing mean rental prices between your tow cities based on the rental data you collected. The focus will be on testing a claim about the difference (or lack of difference) in average rent for 1-bedroom apartments.
Using a 0.05 significance level, you’ll construct your null and alternative hypotheses, perform a two-sample t-test for the means, and interpret your results in plain language. You’ll also explore the potential impact of Type I and Type II errors and reflect on how to strengthen your study.
Objectives
- Make inferences about one or more population means using sample data
- Conduct a two-sample hypothesis test comparing population means
- Use technology (e.g., Excel, calculators, or online tools) to perform statistical tests
- Interpret results in context, including discussion of potential errors
Assignment Steps
- Make a claim based on your observations:
- Do you expect Lakewood's average rent to be higher, lower, or about the same as Golden’s?
- Write both your null hypothesis (H₀) and your alternative hypothesis (H₁) using proper statistical notation. For example:
H₀: μLakewood = μGolden
H₁: μLakewood ≠ μGolden(two-tailed) or use>/<for one-tailed tests.
- Conduct a two-sample t-test at the 0.05 significance level:
- Calculate or report your sample means, standard deviations, sample sizes, and test statistic
- Compare your test statistic with a critical value or use a p-value to make your conclusion
- Clearly state the conclusion in plain language (avoid technical phrases like “fail to reject H₀”)
- Discuss Type I and Type II errors:
- Describe each type of error in your context (e.g., assuming there’s a difference in rent when there isn’t — or vice versa)
- What are the real-world consequences if an error occurs? Consider stakeholders such as community leaders, renters, or housing advocates.
- Reflect on your study design:
- Would having more data help your conclusion?
- Could a different sampling method improve your results?
- What would help you be more confident in your findings?
What to Include in Your Final Paper (Draft Version)
This assignment builds a section of your final project paper titled: Hypothesis Test: Comparing City Rental Prices. Your write-up should include:
- The original claim (clearly stated)
- Null and alternative hypotheses, sample data from both cities
- Test results: test statistic, degrees of freedom, p-value or critical value, and plain language conclusion
- Real-world meaning of the findings: who it affects and how
- Discussion of potential errors and what could improve confidence
Helpful Tips
- Each sample should contain 40 listings — don’t combine them into one group
- You can use a calculator, worksheet, or tech tool for test statistic and p-value
- Stick with a two-sample t-test assuming unequal variances unless instructed otherwise
- Include unit labels (e.g., rent in dollars) in your explanation
- Use clear, everyday language to help your reader (and yourself!) explain what happened in the test
Submission Instructions
- Submit your work as a single Excel workbook or Word document
- Include the hypothesis test setup, calculations, and interpretation
- File name format:
LastName_HypothesisTest_Lakewood_Golden - Upload to the “Project Assignments” folder in our LMS