Deliverable 05 Worksheet Instructions: The Following 034269

Deliverable 05 Worksheet instructions: The Following Worksheet Describ

The worksheet involves analyzing two statistical scenarios: one for independent samples and the other for dependent samples. For each case, you are required to formulate hypotheses, determine the nature of the test (one-tailed or two-tailed), compute the relevant test statistic and critical value or P-value, and then make informed decisions about the null hypotheses. All explanations should be detailed and in your own words, demonstrating a clear understanding of the steps involved in hypothesis testing for both independent and dependent samples, using significance levels of 0.01 and 0.05 accordingly.

Paper For Above instruction

Hypothesis testing is a fundamental aspect of statistical analysis that allows researchers to make decisions about population parameters based on sample data. The scenario provided involves two separate experiments to evaluate the effectiveness of a new drug in reducing blood pressure, one involving independent samples (different groups) and the other involving dependent samples (paired measurements within the same group). Carefully approaching each case requires understanding the appropriate hypotheses, the nature of the test, calculations of test statistics, and interpreting results in context.

Independent Samples: Testing the Effect of a New Drug on Blood Pressure

In this scenario, researchers seek to determine if the new drug effectively lowers systolic blood pressure in individuals with hypertension. Two independent groups—one receiving the treatment and the other serving as a control—are compared based on their mean blood pressures. Since the population standard deviations are unknown, a t-test for independent samples is appropriate. The hypotheses in symbolic form are:

• Null hypothesis (H₀): μ₁ ≥ μ₂ (the treatment does not reduce or potentially increases blood pressure)
• Alternative hypothesis (H₁): μ₁

This is a left-tailed test because we're testing if the treatment group’s mean is less than the control group’s mean. The rationale is based on the claim that the drug should lower blood pressure, which corresponds to the mean being less in the treatment group.

Next, the critical value for the t-distribution at α = 0.01 (significance level) is determined based on degrees of freedom approximated via the Welch-Satterthwaite equation, considering the sample sizes and sample standard deviations. The test statistic is computed using the sample means, sample standard deviations, and sizes. Then, the calculated t-value is compared to the critical t-value to decide whether to reject H₀.

If the test statistic exceeds the critical value (since it is a left-tailed test, we look for t

In this specific scenario, calculations are performed to find the t-statistic and the critical value, and interpret the outcome accordingly. If H₀ is rejected, it supports the claim that the drug effectively lowers blood pressure.

Dependent Samples: Paired Testing Before and After Drug Administration

The second experiment involves measuring blood pressure in the same subjects before and after administering the drug, leading to paired or dependent samples. Here, the hypothesis tests whether the mean difference (before - after) exceeds zero, which would suggest the drug lowers blood pressure.

Formally, the hypotheses are:

• H₀: μ_d ≤ 0 (the mean difference is zero or less, indicating no decrease or increase)
• H₁: μ_d > 0 (the mean difference is greater than zero, indicating a reduction in blood pressure)

This is a right-tailed test because the claim is that the drug helps lower blood pressure, implying the difference (before - after) should be positive.

Calculations involve determining the mean and standard deviation of the differences, computing the t-statistic by dividing the mean difference by the standard error, and then determining the P-value associated with this t-statistic. If the P-value is less than the significance level of 0.05, then H₀ is rejected.

The conclusion in nontechnical terms centers around whether there is sufficient evidence to say that the drug causes a significant reduction in blood pressure for the population, based on the paired sample data.

Conclusion

In both scenarios, careful statistical reasoning enables researchers to draw meaningful conclusions about the drug’s efficacy. The independent samples test assesses whether blood pressure differs significantly between treated and untreated groups, while the dependent samples test evaluates whether blood pressure measurements decrease following treatment within the same group. Proper hypothesis formulation, test selection, calculation, and interpretation are essential for valid inference. If the tests show statistically significant results, they support the drug’s effectiveness in lowering blood pressure; if not, the evidence is insufficient to confirm such a claim.

References

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