Suppose The Manufacturer Of Advil, A Common Headache Remedy

Suppose The Manufacturer Of Advil A Common Headache Remedy Recen

Suppose The Manufacturer Of Advil A Common Headache Remedy Recen

Solve the following statistical problems based on the scenarios provided:

1. In the first scenario, a new formulation of Advil is tested for effectiveness. A sample of 270 current users tried the new drug, with 251 indicating it was more effective. Simultaneously, a control group of 390 users was told the drug was a new formulation, and 352 claimed it improved. State the decision rule for testing at a significance level of 0.01, with hypotheses:

  • H₀: π_n ≤ π_c
  • H₁: π_n > π_c

Rearranged, this means testing whether the proportion of users finding the new drug effective in the trial exceeds the proportion in the control group.

2. Calculate the test statistic for the difference in proportions based on the data provided.

3. Based on the calculations, determine whether to reject H₀ at the 0.01 significance level, concluding if the new drug is more effective.

4. In the second scenario, a study measures attention span (in seconds) of 12-year-old children exposed to three types of commercials: clothes, food, and toys. Using ANOVA techniques, complete the ANOVA table at a significance level of 0.05, including calculating sums of squares (SS), mean squares (MS), and the F-statistic.

5. Using the data, find the mean and standard deviation for each group (clothes, food, and toys), rounding to three decimal places.

6. Test whether there are any significant differences in the mean attention spans across the three types of commercials at the 0.05 significance level. State the null and alternative hypotheses and interpret the results.

7. If the overall ANOVA shows significant differences, perform pairwise comparisons to identify which treatment means differ significantly. Specifically, examine whether treatment groups have different mean attention spans at a 95% confidence level.

8. Complete the calculations for sums of squares (SST, SSE, SS total), the ANOVA table, and make a decision regarding the null hypothesis. If H₀ is rejected, conclude whether treatments 1 and 2 (or the different commercial types) have significantly different effects on attention span.

Paper For Above instruction

The evaluation of new pharmaceutical formulations, such as the recent development of an improved Advil, necessitates rigorous statistical analysis to determine efficacy differences. In analyzing whether the new formulation is statistically more effective than the existing one, hypothesis testing for proportions is applied. The primary hypotheses are: H₀: π_n ≤ π_c (the new formulation's effectiveness proportion does not exceed that of the current formulation), against H₁: π_n > π_c (the new formulation is more effective). Given the data—270 trial participants with 251 perceiving improvement, and a control group of 390 with 352 perceiving improvement—the initial step involves computing the test statistic for the difference in sample proportions and establishing the decision rule at a 0.01 significance level.

Calculating the sample proportions, p̂₁ = 251/270 ≈ 0.9296 and p̂₂ = 352/390 ≈ 0.9026, the pooled proportion p̂_p = (251 + 352) / (270 + 390) = 603/660 ≈ 0.9136. The standard error (SE) of the difference between proportions is derived from the pooled proportion: SE = sqrt [p̂_p(1 - p̂_p)(1/270 + 1/390)] ≈ 0.0298. The test statistic (z) is then calculated as (p̂₁ - p̂₂) / SE ≈ (0.9296 - 0.9026) / 0.0298 ≈ 0.9276. For a significance level of 0.01, the critical z-value for a one-tailed test is approximately 2.33. Since 0.9276

In the second part, analyzing the attention span data involves an ANOVA test to determine whether the mean attention spans differ significantly among children exposed to clothes, food, and toy commercials. The initial step involves calculating the group means and standard deviations for each category: for example, suppose the mean attention span for clothes ads is M₁, food ads M₂, and toys M₃, with respective sample sizes and standard deviations reported. The ANOVA table consists of sums of squares: total SS, treatments SS (between-group variability), and error SS (within-group variability), with degrees of freedom accordingly. The mean squares are obtained by dividing SS by their respective degrees of freedom, and the F-statistic is computed as MS_treatments / MS_error.

An F-test at a 0.05 significance level assesses whether the observed between-group variability exceeds what would be expected by chance under the null hypothesis of equal means (H₀: μ₁ = μ₂ = μ₃). If the calculated F exceeds the critical F-value, we reject H₀ and conclude that at least one commercial type significantly affects attention span differently. Follow-up pairwise comparisons, such as Tukey's HSD, help identify specific pairs with significant differences. The results indicate whether the type of advertisement influences attention span in a statistically meaningful manner, bearing implications for targeted advertising strategies and pediatric attention research.

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