10.1.4: Interval Estimates
- Page ID
- 63625
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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 4: Confidence Interval Analysis (Comparing Two Cities)
In this portion of your semester project, you will apply the concepts of confidence intervals to the data you collected for two cities (40 listings from each). Confidence intervals help us estimate unknown population parameters, like the true average rent or the proportion of listings with a feature like pet-friendly policies, based on sample data.
You’ll compute confidence intervals for the mean rental price of 1-bedroom apartments in each city at the 80% and 99% confidence levels. You'll also estimate a population proportion (e.g., % of pet-friendly listings) for each city using your qualitative variable. Finally, you’ll compare the city results and reflect on how different confidence levels affect interpretation.
Objectives
- Make inferences about two populations using sample data
- Construct confidence intervals for means and proportions
- Compare intervals across different confidence levels
- Interpret results in real-world language and context
Assignment Steps
- Estimate the mean rental price for each city at both the 80% and 99% confidence levels. Use the confidence interval formula, your sample mean, sample standard deviation, sample size, and critical t* value. Show all work clearly.
- Compare the confidence intervals across the two cities:
- Do the intervals overlap?
- What does each interval tell you about the city's rental market?
- Discuss the impact of using a wider (99%) vs. narrower (80%) confidence level
- Estimate a proportion from each city for your qualitative variable (e.g., % of listings that are pet-friendly).
- Select a reasonable confidence level (90%, 95%, or 99%)
- Use the formula for a confidence interval for proportions
- Report each sample proportion and the corresponding margin of error
- Interpret and reflect:
- What conclusions can you draw about rental markets across the two cities?
- What trends do the confidence intervals suggest?
- In what situations would a lower vs. higher confidence level be most appropriate?
- Final reflection: Who might benefit from these insights? (e.g., renters, landlords, city planners)
What to Include in Your Final Paper (Draft Version)
This section becomes part of your project report under: Comparing Estimates from Two Cities. Your draft should include:
- Listed confidence intervals (including level used)
- Visual aids (if helpful) comparing intervals side by side
- Interpretation of overlap or separation in values
- Brief conclusion: what can or cannot be said about the differences in rental price and proportions between the two places?
Helpful Tips
- State the cities clearly so that your intervals don’t get mixed up!
- Remember: the sample size for each city is 40, so n = 40 for each set of calculations
- You won’t combine the samples, treat them separately first
- Use a consistent qualitative variable across both datasets (e.g., parking type for both cities)
- When comparing proportions, think in terms of what seems "significantly different" vs. roughly the same
Submission Instructions
- Submit a single Excel file or document including:
- Calculated mean and proportion confidence intervals for both cities
- Any necessary tables or labeled charts
- Written comparison and reflection
- File name format:
LastName_CIComparison.xlsx