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Determine whether the samples are independent or dependent. The effectiveness of a new headache medicine is tested by measuring the amount of time before the headache is cured for patients who use the medicine and another group of patients who use a placebo drug. Please explain your decision.

Determine whether the samples are independent or dependent. The effectiveness of a headache medicine is tested by measuring the intensity of a headache in patients before and after drug treatment. The data consist of before and after intensities for each patient. Please explain your decision.

In a random sample of 500 people aged 20-24, 22% were smokers. In a random sample of 450 people aged 25-29, 14% were smokers. Test the claim that the proportion of smokers in the two age groups is the same. Use a significance level of 0.01. (Show all steps of the hypothesis test and all calculations)

A researcher wishes to determine whether people with high blood pressure can reduce their blood pressure by following a particular diet. The sample data is shown below, where μ₁ represents the mean blood pressure of the treatment group and μ₂ represents the mean for the control group. Use a significance level of 0.01 and whichever method you deem appropriate (p-value method, critical value method, or confidence interval method) to test the claim that the diet reduces the blood pressure. We do not know the values of the population standard deviations. Use Microsoft Excel or the following t-distribution table: Treatment Group Control Group n₁=85, n₂=203, s₁=38.7, s₂=39.2. Instructor Comments:

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Determining whether samples are independent or dependent is fundamental in statistical analysis as it influences the choice of the appropriate test to analyze data accurately. The nature of the specific study design informs this determination. The first scenario involves testing the effectiveness of a new headache medicine by measuring the time to relief in two distinct groups — those using the medicine and those using a placebo. Since these two groups consist of different patients, their data are collected independently; each patient in one group has no direct pairing with any patient in the other group. Therefore, the samples are independent. This scenario calls for an independent samples t-test, which compares the mean response times between two independent groups, assuming the data meet the necessary assumptions of normality or large sample size justification, and equal variances if applicable.

The second scenario examines the same patients before and after drug treatment by measuring the headache intensity. Here, the data are paired because each patient serves as their own control. The before-and-after measurements are related, as they come from the same individual, which introduces dependence. Analyzing these paired data typically involves a paired t-test, which compares mean differences within subjects. This approach accounts for the relatedness of the measurements, improving the test's power and accuracy, plus controlling for individual variability that could confound the results if treated as independent samples.

In the case of comparing smoking proportions among different age groups, the data are from two independent samples: one from individuals aged 20-24 and another from those aged 25-29. These samples are unrelated, and the analysis involves testing the difference between two proportions. The appropriate statistical test is the two-proportion z-test, which evaluates whether the observed difference in smoking percentages is statistically significant at a 0.01 significance level. This test assumes independent samples, random sampling, and sufficient sample sizes for the normal approximation to hold.

The final example considers whether a diet reduces blood pressure among individuals with hypertension, where the sample involves two groups: a treatment group and a control group. The data clarify that these are independent samples since the individuals are different in each group. The objective is to compare the mean blood pressures using a two-sample t-test. Since the population standard deviations are unknown, the t-test with degrees of freedom calculated via the Satterthwaite approximation is appropriate. This analysis helps determine whether the observed difference in mean blood pressure reductions is statistically significant at the 0.01 level, providing evidence for or against the diet’s efficacy.

In summary, understanding whether samples are independent or dependent hinges on the study design. Independent samples come from unrelated groups, suitable for tests comparing means or proportions between two groups. Dependent samples involve paired data, such as repeated measures from the same subjects, and require paired tests to account for the relatedness. Correctly identifying this distinction ensures that the appropriate statistical methodology is applied, leading to valid and reliable inferences in research.

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