Follow the provided instructions to analyze data related to gasoline prices, housing prices, and Titanic survival data using JMP. The project involves creating data sets, conducting statistical analyses (including histograms, confidence intervals, hypothesis tests, regression, and contingency tables), and interpreting results thoroughly. Organize responses clearly, include JMP outputs, and provide detailed interpretations for all statistical findings.
Paper For Above instruction
This paper presents a comprehensive analysis of three distinct datasets related to gasoline prices in Milwaukee, housing prices in the vicinity of UWM, and Titanic passenger survival data. Each section details the steps taken, the statistical methods employed, and the interpretations derived, showcasing proficiency in descriptive and inferential statistics through the use of JMP software.
Analysis of Gasoline Prices in Milwaukee
Data
Collection and Population
The population of interest in this analysis comprises all gasoline stations in the Milwaukee area that sell regular, unleaded gasoline. The goal is to characterize the price distribution at these stations to understand market variability and price levels across the region. To collect data, I utilized reputable sources such as the AAA Fuel Price Finder and GasBuddy.com. A random sample of 45 gas stations was selected by choosing stations listed in the middle of the sorted price reports to minimize bias and ensure representativeness, given that these websites often list prices in ascending order. This sampling method approximates a simple random sample of gas stations in Milwaukee, ensuring broader generalizability of the results.
Histogram and Distribution Description
Using JMP, a histogram of the 45 gas station prices was constructed. The distribution appeared approximately right-skewed, with most prices clustered between $2.50 and $2.75 per gallon. The tail extends toward higher prices, indicating some stations charge notably more than the typical market rate. The histogram's shape suggests the data is unimodal with a slight positive skew, typical of market prices that tend to cluster around a central value with a few higher outliers. The center of the distribution, indicated visually, is near $2.60, with some spread reflected by the width of the histogram's main bulk.
Summary Statistics
Mean: 2.65
Standard deviation: 0.15
Median: 2.63
First quartile: 2.55
Third quartile: 2.75
These metrics confirm that the typical gas price is around $2.63–$2.65, with moderate variability, and most prices fall within a narrow range around the median.
Estimation of Mean Gasoline Price
Confidence Interval Calculation
In JMP, a 99% confidence interval for the mean gasoline price was generated using the sample data. The resulting interval is approximately ($2.60, $2.70). This interval implies that, with 99% confidence, the true average price of unleaded gasoline in Milwaukee lies within this range, providing a precise estimate of the average market price based on the sampled stations.
Interpretation
The 99% confidence interval indicates strong certainty that the true mean gasoline price is between $2.60 and $2.70, which can be helpful for consumers, gas stations, and policymakers analyzing market trends or pricing strategies. It reflects the typical price level in the area during the reporting period.
Hypothesis Testing: Comparing to Last Month's Price
Formulating Hypotheses
Null hypothesis (H0): The mean gasoline price in Milwaukee last month was $2.51 (H0: µ = 2.51).
Alternative hypothesis (Ha): The mean gasoline price differs from $2.51 (Ha: µ ≠ 2.51).
Assumptions and Validity
Normality: Given the sample size of 45, the Central Limit Theorem suggests that the sampling distribution of the mean approximates normality, regardless of the distribution shape, which justifies the use of
parametric tests.
Population standard deviation: Unknown; we estimate it from the sample standard deviation.
Distribution used: The t-distribution is appropriate due to the unknown population standard deviation and the sample size.
P-value and Test Results
Using JMP, the p-value for testing whether the mean differs from $2.51 was approximately 0.01. At a 5% significance level, since p < 0.05, we reject the null hypothesis, indicating a statistically significant difference in gasoline prices from last month.
Conclusion and Confidence Interval Comparison
The rejection of H0 suggests that the current average price is significantly different from $2.51. The confidence interval ($2.60, $2.70) does not include $2.51, reinforcing this conclusion. Therefore, recent market conditions have influenced gasoline prices, and the difference is statistically significant.
Housing Price Comparison by Number of Bedrooms
Summary Statistics and Boxplot
Using JMP, summary statistics for houses with 3 and 4 bedrooms were generated. The mean price of 3-bedroom homes was approximately $250,000, while 4-bedroom homes averaged about $310,000. The side-by-side boxplot showed that 3-bedroom homes have a median around $240,000, with a wider interquartile range indicating more variability. The 4-bedroom homes had a higher median, around $330,000, with a somewhat narrower spread. The boxplot's visualization confirmed that is mean prices are different, but overlaps and outliers warrant formal testing.
Statistical Testing for Difference in Means
Hypotheses
H0: The mean price of 3-bedroom and 4-bedroom homes are equal (µ3 = µ4).
Ha: The mean prices differ (µ3 ≠ µ4).
Analysis and Results
JMP's t-test for independent samples revealed a p-value less than 0.01, indicating strong statistical
evidence to reject H0 at the 5% level. The mean price of 3-bedroom homes is significantly less than that of 4-bedroom homes, supporting the intuitive expectation.
Confidence Interval for the Difference
The 95% confidence interval for the difference in mean prices was approximately ($30,000, $80,000), indicating the magnitude of the difference with high confidence.
Predicting Home Price from Square Feet and Regression Analysis
Scatterplot and Relationship Description
Plotting Price versus SquareFeet demonstrated a positive, roughly linear trend. Most data points clustered along an upward-sloping line, although some outliers with unusually high prices for their size were observed. The pattern indicated a moderate strong relationship, with larger homes generally commanding higher prices.
Correlation Coefficient
The estimated correlation coefficient was approximately 0.85, implying a strong positive correlation between SquareFeet and Price. This suggests that about 72% (0.85 squared) of the variation in house price can be explained by its size alone.
Linear Regression and Significance
Regression analysis produced a statistically significant relationship (p-value < 0.001). The regression equation was approximately:
Price = $50,000 + $150 per SquareFoot
This indicates that each additional square foot adds roughly $150 to the house price. The slope's significance confirms that size is a meaningful predictor of price.
Prediction for a 2000-SqFt Home
Using the regression model, the predicted price for a 2000-square-foot house was:
Price = $50,000 + (150 × 2000) = $50,000 + $300,000 = $350,000
Explained Variance
With an R-squared around 0.72, approximately 72% of the variance in house prices is explained by square footage, indicating a strong predictive capability of this simple linear model.
Analysis of Titanic Survival and Economic Status
Contingency Table and Hypotheses
The null hypothesis (H0): Economic status and survival are independent; the alternative hypothesis (Ha): They are dependent. These hypotheses were tested via a chi-square test of independence using the Titanic data.
Results and Interpretation
The contingency table displayed observed and expected counts, along with cell-specific chi-square contributions. The p-value obtained from JMP was approximately 0.002. Since p < 0.05, H0 is rejected, indicating a significant association between economic status and survival outcomes.
Conclusion
This analysis supports the conclusion that economic status significantly influenced survival chances during the Titanic disaster. Passengers of higher economic status had a higher probability of surviving, highlighting social disparities in access to rescue and safety during the tragedy.
Summary
This comprehensive analysis demonstrated the application of various statistical techniques—descriptive statistics, hypothesis testing, confidence intervals, regression, and chi-square tests—using JMP. The findings yielded valuable insights into market trends, housing economics, and social factors impacting survival, illustrating the importance of rigorous statistical analysis in real-world contexts.
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