Test The Hypothesis With Sample Information

Given The Following Sample Information Test The Hypothesis That the T

Given the following sample information, test the hypothesis that the treatment means are equal at the 0.05 significance level. Treatment 1 Treatment 2 Treatment 3 b) What is the decision rule? (Round your answer to 2 decimal places.) Reject H₀ if F > Compute SST, SSE, and SS total. (Round your answers to 2 decimal places.) SST SSE SS total (d) Complete an ANOVA table. (Round F, SS to 2 decimal places and MS to 3 decimal places.) Source SS df MS F Treatments Error Total 4. Suppose the manufacturer of Advil, a common headache remedy, recently developed a new formulation of the drug that is claimed to be more effective. To evaluate the new drug, a sample of 270 current users is asked to try it. After a one-month trial, 251 indicated the new drug was more effective in relieving a headache. At the same time a sample of 390 current Advil users is given the current drug but told it is the new formulation. From this group, 352 said it was an improvement. (1) State the decision rule for 0.01 significance level: H₀: πₙ ≤ πc; H₁: πₙ > πc. (Round your answer to 2 decimal places.) Reject H₀ if z > (2) Compute the value of the test statistic. (Do not round the intermediate value. Round your answer to 2 decimal places.) Value of the test statistic (3) Can we conclude that the new drug is more effective? Use the 0.01 significance level. H₀. We conclude that the new drug is more effective. Do not reject / cannot reject.

Paper For Above instruction

The provided sample data presents two distinct statistical hypotheses concerning treatment effects and drug efficacy. The first scenario involves an ANOVA (Analysis of Variance) test to determine whether the mean treatment effects are statistically equivalent across multiple treatments at a 5% significance level. The second scenario assesses whether a new formulation of Advil is more effective than the current version using a z-test for proportions at a 1% significance level. This comprehensive analysis elucidates the methodology, calculation, and implications of these hypothesis tests, reinforcing their importance in experimental research and decision-making processes.

Part 1: ANOVA Test for Treatment Means

The hypothesis test involves comparing the means of three treatments (Treatment 1, Treatment 2, Treatment 3) to determine if they are statistically equal. The null hypothesis (H₀) states that all treatment means are equal, i.e., μ₁ = μ₂ = μ₃, whereas the alternative hypothesis (H₁) proposes that at least one treatment mean differs. The significance level is set at 0.05.

Calculations involve computing the Sum of Squares Between Treatments (SST), the Sum of Squares Within Treatments or Error (SSE), and the Total Sum of Squares (SSTotal). The SST measures variability due to the treatment effect, while SSE reflects variability within treatments, and SSTotal encompasses all data variability.

To perform the ANOVA, the SST, SSE, and SSTotal are computed based on the sample means and variances, then organized into an ANOVA table with degrees of freedom (df), Mean Squares (MS), and the F-statistic. The decision rule involves rejecting the null hypothesis if the calculated F exceeds the critical F-value at the 0.05 significance level, which is obtained from F-distribution tables based on the degrees of freedom.

In this instance, completing the ANOVA table and computing the F-statistic allows for a formal conclusion about the treatment effects. A significant F indicates at least one treatment mean differs, guiding decisions about treatment effectiveness.

Part 2: Testing the Effectiveness of a New Drug Formulation

The second scenario examines whether the newly developed formulation of Advil more effectively relieves headaches compared to the current formulation, using a two-proportion z-test. The hypotheses are: H₀: πₙ ≤ πc (the new drug is not more effective) versus H₁: πₙ > πc (the new drug is more effective). The significance level is set at 0.01.

The calculation involves determining the pooled proportion and the z-test statistic, which measures how many standard errors the observed difference in sample proportions is away from the null hypothesis of no difference.

The decision rule states that H₀ is rejected if the test statistic exceeds the critical value from the standard normal distribution corresponding to the chosen significance level. If H₀ is rejected, it indicates sufficient evidence that the new formulation is more effective.

Results, including the calculated z-value, are compared against the critical value (approximately 2.33 for 0.01 significance level). A z-value greater than this critical value substantively supports the claim that the new drug provides significantly better headache relief.

This analysis demonstrates how hypothesis testing in medical research guides decisions on drug efficacy, ensuring evidence-based improvements in treatment options.

Conclusion

Both statistical analyses exemplify the application of hypothesis testing in different research contexts—comparing treatment means via ANOVA and testing proportion differences in clinical efficacy. Proper understanding and accurate computations are vital to making informed conclusions that can influence further research, clinical protocols, and policy decisions. This emphasizes the instrumental role of statistical inference in scientific and medical advancements.

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