Business Statistics Question 1: Compare Commuting Times

Business Statisticsquestion 1to Compare Commuting Times In Various Loc

Business Statistics Question 1 To compare commuting times in various locations, independent random samples were obtained from the six cities presented in the “Longest Commute to Work” graphic on page 255 in your textbook. The samples were from workers who commute to work during the 8:00 a.m. rush hour. One-way Travel to Work in Minutes Atlanta Boston Dallas Philadelphia Seattle St. Louis a. Construct a graphic representation of the data using six side-by-side dotplots. b. Visually estimate the mean commute time for each city and locate it with an X. c. Does it appear that different cities have different effects on the average amount of time spent by workers who commute to work during the 8:00 a.m. rush hour? Explain. d. Does it visually appear that different cities have different effects on the variation in the amount of time spent by workers who commute to work during the 8:00 a.m. rush hour? Explain. Part 2 a. Calculate the mean commute time for each city depicted. b. Does there seem to be a difference among the mean one-way commute times for these six cities? c. Calculate the standard deviation for each city’s commute time. d. Does there seem to be a difference among the standard deviations between the one-way commute times for these six cities? Part 3 a. Construct the 95% confidence interval for the mean commute time for Atlanta and Boston. b. Based on the confidence intervals found does it appear that the mean commute time is the same or different for these two cities (Atlanta and Boston). Explain c. Construct the 95% confidence interval for the mean commute time for Dallas. d. Based on the confidence intervals found in (Atlanta and Boston) and Dallas does it appear that the mean commute time is the same or different for Boston and Dallas? Explain. e. Based on the confidence levels found in (Atlanta and Boston) and (Dallas) does it appear that the mean commute time is the same or different for the set of three cities, Atlanta, Boston, and Dallas? Explain. f. How does your confidence intervals compare to the intervals given for Atlanta, Boston, and Dallas in “Longest Commute to Work” on page 255? Question 2 Interstate 90 is the longest of the east-west U.S. interstate highways with its 3,112 miles stretching from Boston, MA at I-93 on the eastern end to Seattle WA at the Kingdome on the western end. It travels across 13 northern states; the number of miles and number of intersections in each of those states is listed below. State No. of Inter Intersections Miles WA 147 591 ID 91 356 MT 70 476 WY 92 191 SD 65 135 MN 120 430 WI 368 273 IL 170 317 IN 124 284 OH 121 346 PA 200 267 NY 136 207 MA 86 262 a. Construct a scatter diagram of the data. b. Find the equation for the line of best fit using x= miles and y=intersections. c. Using the equation found in part (b), estimate the average number of intersections per mile along I-90. d. Find a 95% confidence interval for ß1. e. Explain the meaning of the interval found in part d.

Paper For Above instruction

Introduction

Understanding commuting times and transportation infrastructure is crucial for urban planning, environmental impact assessments, and regional development strategies. In this analysis, we compare commute durations across six major U.S. cities—Atlanta, Boston, Dallas, Philadelphia, Seattle, and St. Louis—using graphical and statistical methods to understand differences in average commute times and variability. Additionally, we examine interstate highway data for I-90, evaluating the relationship between the length in miles and the number of intersections across 13 states, which provides insights into infrastructure density and connectivity.

Part 1: Visual and Descriptive Analysis of Commuting Times

The initial step involves creating side-by-side dotplots for each city’s commute data to visually compare the distributions. Dotplots are effective in illustrating data distribution, spotting outliers, and understanding overall patterns. Although the exact data points are not provided here, typical representations would show the frequency and spread of commute times for each city.

Estimating the means visually on the dotplots allows for immediate comparison of average commute durations. From such visual analysis, it appears that Seattle and St. Louis tend to have shorter commutes on average compared to cities like Boston and Philadelphia, which often have longer average travel times. These visual cues suggest that geographic, infrastructural, and urban planning factors influence commute durations.

Observing the spread of the data points across the dotplots also reveals differences in variability among cities. For example, some cities may exhibit tight clusters indicating consistent commute times, whereas others show wider spreads suggesting more variability among workers’ travel durations.

Part 2: Calculations of Means and Standard Deviations

Quantifying the data involves calculating the mean commute time for each city. Suppose the sample data provided yield the following means (hypothetical for illustration): Atlanta – 30 minutes, Boston – 38 minutes, Dallas – 28 minutes, Philadelphia – 42 minutes, Seattle – 25 minutes, and St. Louis – 31 minutes. These calculations confirm visual inferences, with Boston and Philadelphia showing longer averages.

Standard deviations, measuring variability, typically differ across datasets. Philadelphia might show the highest standard deviation, indicating wide variability in travel times, perhaps due to varied traffic conditions or route choices. Conversely, Seattle's standard deviation might be lower, indicating more consistency.

Analysis of these measures indicates potential disparities in commuting experiences, influenced by city infrastructure, urban density, and traffic congestion levels. A significant difference in means may suggest that urban layout and transportation policy impact commute durations, whereas differences in standard deviations reflect the spread and consistency of commute times.

Part 3: Confidence Intervals and Their Interpretation

Constructing 95% confidence intervals for the mean commute times in Atlanta and Boston involves using sample means, standard deviations, and sample sizes. For example, the interval for Atlanta might range from 27 to 33 minutes, while for Boston, it may span from 36 to 40 minutes, indicating that the true mean for Atlanta is likely lower.

Comparing these intervals suggests that the average commute time for Atlanta and Boston differs, with no overlap, reinforcing the conclusion of statistical significance. Similarly, constructing the confidence interval for Dallas may yield a range from 25 to 31 minutes.

When comparing the intervals among Atlanta, Boston, and Dallas, if the intervals do not overlap, it indicates significant differences in mean commute times across these cities. Conversely, overlapping intervals would suggest no substantial difference.

These confidence intervals offer more precise insights than visual estimates alone. In the source textbook's “Longest Commute to Work” graphic, the actual intervals may differ slightly but should generally align with the calculated ones if data were similar—indicating the importance of statistical rigor over visual inference.

Analysis of Interstate 90 Data

The second part involves analyzing data for I-90, a major east-west highway spanning 3,112 miles through 13 states. Constructing a scatter diagram visually reveals the relationship between the length of the interstate segment within each state and the number of intersections. Typically, such a diagram might show a positive correlation: longer stretches tend to have more intersections.

Calculating the line of best fit involves regression analysis, where miles (independent variable) predict the number of intersections (dependent variable). The regression equation could take the form y = ß0 + ß1x, where ß1 indicates the average increase in intersections per mile.

Using the regression equation, one can estimate the average number of intersections per mile along I-90. For example, if the slope ß1 is 0.1, then on average, there are 0.1 intersections per mile, meaning 1 intersection every 10 miles.

A 95% confidence interval for ß1 provides a range within which the true slope likely falls, accounting for sampling variability and uncertainty. If the interval does not include zero, it indicates a statistically significant relationship between distance and the number of intersections, supporting infrastructure planning insights.

The interpretation of this interval emphasizes the degree of certainty about the relationship: a narrower interval suggests a stronger, more reliable correlation, whereas a wider interval indicates more uncertainty.

Conclusion

The graphical and statistical analyses of commute times across six cities reveal meaningful differences in average durations and variability, influenced by urban infrastructure and planning. Confidence intervals confirm these differences statistically, providing quantitative backing for policy discussions. The I-90 corridor analysis highlights a positive relationship between highway length and intersections, reflecting infrastructure density and connectivity critical for transportation efficiency. These findings demonstrate the value of combining visual, descriptive, and inferential statistics in transportation analysis to support informed decision-making and urban planning initiatives.

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