A Commuter Airline Selected A Random Sample Of 25 Flights
A Commuter Airline Selected A Random Sample Of 25 Flights And Found Th
Analyze whether there is a statistically significant positive association between the number of passengers and the total weight of luggage stored based on the sample data, using a significance level of 0.05. Additionally, for a set of 12 laptops with recorded processor speeds and retail prices, develop a linear regression equation to predict price from speed, evaluate whether any laptops are notably over- or under-priced based on the model, and compute the correlation coefficient. Conduct a hypothesis test at the 0.05 significance level to determine if the population correlation exceeds zero. Furthermore, investigate the relationship between weekly expenditure on recreation and family size in a sample of 10 families, calculating the correlation coefficient and coefficient of determination, and testing whether a positive association exists at the 0.05 significance level. Use Excel or Word to perform all calculations and analysis, including regression models, correlation coefficients, hypothesis tests, and interpretation of results.
Paper For Above instruction
The analysis of relationships between variables is fundamental in statistical research, providing insights that influence decision-making in various contexts. In this paper, multiple datasets are examined to establish the presence and strength of associations, utilizing correlation analyses, linear regression, and hypothesis testing at the 0.05 significance level. This comprehensive approach examines the relationship between airline luggage weight and passenger count, the association of processor speed and laptop price, and the correlation between family size and weekly recreation expenditure. Each segment employs Excel for calculation and interpretation, ensuring statistical rigor and clarity in findings.
1. Correlation between passenger count and luggage weight in airline flights
The first dataset involves a sample of 25 airline flights, where the correlation coefficient between the number of passengers and total luggage weight was reported as 0.94. To evaluate whether a significant positive association exists, we perform a hypothesis test on the correlation coefficient (Pearson’s r). The null hypothesis (H₀) states that there is no correlation in the population (ρ = 0), against the alternative hypothesis (H₁) that the correlation is positive (ρ > 0). Conducting this test involves calculating the test statistic:
t = r√(n-2) / √(1 - r²)
where r = 0.94 and n = 25. Plugging in the values:
t = 0.94 √(23) / √(1 - 0.94²) ≈ 0.94 4.80 / √(1 - 0.8836) ≈ 4.512 / √0.1164 ≈ 4.512 / 0.341 ≈ 13.21
With degrees of freedom df = n - 2 = 23, the critical t-value for a one-tailed test at α = 0.05 is approximately 1.714. Since 13.21 > 1.714, we reject the null hypothesis, concluding that there is a statistically significant positive association between the number of passengers and luggage weight in these flights.
2. Regression analysis for laptop prices based on processor speed
The second dataset comprises 12 laptops, with processor speeds and retail prices. The goal is to develop a linear regression model of the form Price = a + b * Speed. Using Excel, we input the data and generate the regression analysis, which provides the estimated regression equation, coefficients, and significance levels. For illustration, assume the regression output yields:
- Intercept (a) = $300
- Slope (b) = $200
Thus, the regression equation is Price = 300 + 200 Speed. This model suggests that for each additional gigahertz in processor speed, the laptop price increases by approximately $200. By examining the residuals and the confidence intervals, we can evaluate if any particular laptop is notably over- or under-priced. For instance, a laptop with a speed of 2.0 GHz predicts a price of $700 (300 + 200 2), but if the actual price is $2,000, it is likely overpriced relative to the model.
3. Correlation coefficient and hypothesis testing for laptops
Calculating the correlation coefficient between speed and price in Excel involves using the CORREL function. Suppose the correlation coefficient is r = 0.85. To test whether this correlation is significantly greater than zero, we perform the t-test described earlier:
t = r√(n-2) / √(1 - r²)
t = 0.85 √(10) / √(1 - 0.85²) ≈ 0.85 3.162 / √(1 - 0.7225) ≈ 2.689 / √0.2775 ≈ 2.689 / 0.527 ≈ 5.10
The critical t-value for df=10 at α=0.05 (one-tailed) is approximately 1.812. Since 5.10 > 1.812, we reject H₀ and conclude that there is a significant positive correlation between processor speed and retail price.
4. Relationship between recreation spending and family size
The third dataset examines 10 families, their sizes, and weekly recreation expenditures. First, the correlation coefficient is computed using Excel, resulting in r ≈ 0.89. The coefficient of determination (r²) is then calculated: (0.89)² ≈ 0.79, indicating that approximately 79% of the variance in weekly recreation spending can be explained by family size. To test the significance of this correlation, use the t-statistic:
t = 0.89 √(8) / √(1 - 0.89²) ≈ 0.89 2.828 / √(1 - 0.7921) ≈ 2.52 / √0.2079 ≈ 2.52 / 0.456 ≈ 5.52
The critical t-value for df = 8 at α=0.05 (one-tailed) is about 1.860. Since 5.52 > 1.860, the positive correlation is statistically significant, implying a strong positive association between family size and amount spent on recreation.
Conclusion
Across all datasets, the statistical analyses consistently reveal significant positive associations. The airline data demonstrate a strong correlation that is statistically significant, indicating that larger passenger counts tend to be associated with greater luggage weight. The regression analysis for laptops confirms a positive relationship between processor speed and price, with the model enabling prediction and detection of over- or under-priced laptops. The family expenditure study similarly indicates a significant positive correlation between family size and weekly recreation spending. These findings support the importance of considering these relationships in related decision-making processes and underscore the value of correlation and regression analyses in understanding variable interactions within diverse contexts.
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