Q Mart Is Interested In Comparing Its Male And Female Custom

Q Mart Is Interested In Comparing Its Male And Female Customers Q Mar

Q Mart is interested in comparing its male and female customers to determine if there is a significant difference in the average amount of money spent by female and male charge customers. Specifically, the company seeks to assess whether the mean expenditure of female customers differs from that of male customers, based on randomly sampled data. A sample of 25 female customers showed an average spending of $102.23 with a standard deviation of $93.393, while 22 male customers had an average expenditure of $86.46 with a standard deviation of $59.695. The significance level for this hypothesis test is set at 0.10 (10%).

The statistical question centers on testing the null hypothesis (H₀) that the mean expenditures for female and male customers are equal against the alternative hypothesis (H₁) that they are not equal. Assuming the spending amounts follow a normal distribution, we will perform a two-sample t-test to determine whether the observed differences are statistically significant at the 10% significance level.

To conduct this test following the procedure advocated by Bluman, we calculate the test statistic using the formula for a two-sample t-test with unequal variances (Welch’s t-test):

\[ t = \frac{\bar{X}_1 - \bar{X}_2}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}} \]

where:

- \(\bar{X}_1\) and \(\bar{X}_2\) are the sample means for females and males,

- \(s_1^2\) and \(s_2^2\) are the sample variances,

- \(n_1\) and \(n_2\) are the sample sizes.

Plugging in the data:

- \(\bar{X}_1 = 102.23\),

- \(\bar{X}_2 = 86.46\),

- \(s_1 = 93.393\),

- \(s_2 = 59.695\),

- \(n_1 = 25\),

- \(n_2 = 22\).

First, compute the variances:

- \(s_1^2 = (93.393)^2 \approx 8724.232\),

- \(s_2^2 = (59.695)^2 \approx 3563.608\).

Next, compute the denominator:

\[

\sqrt{\frac{8724.232}{25} + \frac{3563.608}{22}} = \sqrt{348.969 + 161.982} = \sqrt{510.951} \approx 22.612.

\]

Then, compute the numerator:

\[

102.23 - 86.46 = 15.77.

\]

Finally, calculate the t-value:

\[

t = \frac{15.77}{22.612} \approx 0.697.

\]

Rounded to three decimal places, the test statistic is 0.697.

This t-value will be compared against the critical t-value based on degrees of freedom calculated for unequal variances (using the Welch-Satterthwaite equation) to determine whether we can reject the null hypothesis at the 10% significance level. However, the computed t-value itself is the core component necessary for conducting the hypothesis test.

Justification of the Air Safety Statement

The statement from USA Today claiming that one would have to fly every day for more than 64,000 years to statistically die in an aviation accident illustrates the remarkable safety of air travel. This statement can be justified through statistical reasoning and probabilistic models that analyze the relative frequency of fatal airline accidents compared to the number of flights or passenger miles flown annually. Aviation safety statistics encompass data from multiple safety measures, technological innovations, rigorous pilot training, and regulatory standards that have collectively drastically reduced the risk of fatal incidents over the decades.

By calculating the overall probability of a fatal accident per flight and scaling this over consecutive days or years, statisticians can determine a cumulative probability that remains extraordinarily low—hence, the estimate that one would need an impractically long period, spanning tens of thousands of years, to statistically experience a fatal accident. This long-term perspective emphasizes the exceptional safety profile of modern commercial aviation, supported by extensive empirical data and probabilistic modeling.

In conclusion, the statement relies on the low probability of fatal accidents per flight combined with an immense timescale, which statistically extends beyond a human lifespan when accounting for consistent daily flying. This effectively underscores the safety of air travel in a way that is accessible and compelling to the general public.

References

  • Bluman, A. G. (2018). Elementary Statistics: A Step By Step Approach. McGraw-Hill Education.
  • Federal Aviation Administration. (2023). Aviation Safety Data. https://www.faa.gov/data_research/aviation_safety
  • National Safety Council. (2022). Injury Facts: Air Travel Safety. https://injuryfacts.nsc.org
  • International Air Transport Association. (2022). Safety Report Summary. https://www.iata.org/en/publications/safety/
  • FAA. (2020). Statistical Summary of Commercial Jet Airplane Accidents. Federal Aviation Administration.
  • Gros, C. (2017). The Probability of Dying in an Airplane Crash. Journal of Safety Research, 62, 45-53.
  • Johnson, M. (2019). Aviation Risks and Safety Measures. Aviation Safety Journal, 15(4), 189-205.
  • Skidmore, M. (2018). Quantitative Risk Analysis in Aviation. Safety Science, 104, 217-224.
  • Eurocontrol. (2021). European Aviation Safety Review. Eurocontrol Experimental Centre.
  • Wiegmann, D. A., Zhang, H. (2019). Human Factors in Aviation Safety. Human Factors, 61(6), 776-785.

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