Comparing BMI by Smoking Status

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This pre-assessment will make use of the BODY DATA Excel file created in ST3001. Comparing BMI by smoking status You are interested in determining if smokers have a BMI that is greater than nonsmokers.

  1. Explain which hypothesis test would be appropriate for this situation (assume population variances are not known and sample variances are not equal).
  2. Explain whether this is a left-tailed right-tailed or two-tailed test and justify your choice.

Comparing BMI by smoking status

To determine whether smokers have a BMI that is greater than nonsmokers, we need to conduct a hypothesis test comparing the means of two independent samples: the BMI of smokers and the BMI of nonsmokers. The test will allow us to evaluate whether there is a significant difference between the BMI of smokers and nonsmokers, particularly focusing on whether smokers tend to have a higher BMI than nonsmokers.

1. Hypothesis Test Selection

In this case, we are comparing the means of two independent groups (smokers and nonsmokers), and we do not know the population variances for BMI. Additionally, we are assuming that the sample variances are unequal. Given these assumptions, the most appropriate hypothesis test for this scenario is the Welch’s t-test.

Welch’s t-test is a variation of the standard two-sample t-test that does not assume equal variances between the two groups. It is specifically designed for cases where the assumption of equal variances is violated, making it suitable when comparing BMI between smokers and nonsmokers if their variances differ.

  • Null Hypothesis (H₀): The mean BMI of smokers is equal to the mean BMI of nonsmokers. Mathematically, this can be written as:

    H0:μsmokers=μnonsmokersH₀: \mu_{\text{smokers}} = \mu_{\text{nonsmokers}}

  • Alternative Hypothesis (H₁): The mean BMI of smokers is greater than the mean BMI of nonsmokers. This can be written as:

    H1:μsmokers>μnonsmokersH₁: \mu_{\text{smokers}} > \mu_{\text{nonsmokers}}

2. Tail of the Test

This scenario describes a situation where we are interested in determining whether smokers have a higher BMI than nonsmokers. Since the focus is on whether the BMI of smokers is greater than that of nonsmokers, this leads us to perform a right-tailed test.

In a right-tailed test, we assess whether the mean of one group is significantly greater than the mean of the other group. If the data shows that the mean BMI of smokers is higher than the mean BMI of nonsmokers, and the test statistic falls into the right-hand tail of the distribution, we can reject the null hypothesis and conclude that smokers have a significantly higher BMI.

  • Justification for a Right-Tailed Test: The alternative hypothesis H1:μsmokers>μnonsmokersH₁: \mu_{\text{smokers}} > \mu_{\text{nonsmokers}} indicates that we are only concerned with whether the BMI of smokers is greater, not just different (which would suggest a two-tailed test). Since we are not investigating whether smokers have a lower BMI, a one-tailed (specifically right-tailed) test is appropriate. This choice is based on the directional nature of our research question.

Conclusion

To summarize, the most appropriate hypothesis test for comparing BMI by smoking status, assuming unequal sample variances, is Welch’s t-test. The hypothesis test will be right-tailed, as we are testing whether smokers have a higher BMI than nonsmokers. The results of the test will determine whether there is statistical evidence to support the claim that smokers tend to have a higher BMI than nonsmokers.

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