Consider The Following Hypothesis The Population

Consider The Following Hypothesesh0 Μ 260ha Μ 260the Popul

Consider the following hypotheses: H0: μ = 260 HA: μ ≠ 260 The population is normally distributed with a population standard deviation of 72. a. Use a 1% level of significance to determine the critical value(s) of the test. b-1. Calculate the value of the test statistic with formula x̄ = 294 and n = 85. b-2. What is the conclusion at α = 0.01? c. Use a 10% level of significance to determine the critical value(s) of the test. d-1. Calculate the value of the test statistic with formula x̄ = 238 and n = 85. d-2. What is the conclusion at α = 0.10? 1 A testing organization is evaluating the effectiveness of sinus relief tablets, Relief and SineOut. As part of the analysis, the organization collects 44 sinus sufferers and randomly assigns them to one of two groups. The first group gets Relief, and the other gets SineOut. The length of time to get relief is measured. The mean time to get relief for the Relief group was 31.0 minutes (s.d = 5.1) and for the SineOut group was 34.3 minutes (s.d. = 5.9). Test the hypothesis that the two medications differ in the time to take effect at the 5% significance level. You may assume that the time to take effect in each population is relatively normal. A. State the null and alternate hypotheses. B. State the test statistic. C. State the decision rule. D. Show the appropriate calculations and state the statistical decision. E. What conclusions can you draw concerning the effectiveness of the medications?

Paper For Above instruction

The given set of problems revolves around hypothesis testing, specifically involving population means, and compares the effectiveness of two medications using statistical methods. These problems are central to inferential statistics, which enable us to make decisions about populations based on sample data. This paper will analyze and solve these problems step-by-step, providing an understanding of hypothesis testing frameworks, critical value determinations, test statistics calculations, and interpretive conclusions.)

Introduction

Hypothesis testing is a fundamental statistical procedure used to evaluate claims or assumptions about a population parameter. It involves formulating null and alternative hypotheses, selecting significance levels, calculating test statistics, and making decisions based on critical values or p-values. Here, we explore two scenarios: one concerning a population mean with a known standard deviation, and another comparing two medication effects through a two-sample t-test.

Part 1: Hypothesis Testing for a Single Population Mean

The first two problems concern determining whether a population mean differs from a hypothesized value (260) using different significance levels. The population is normally distributed with a known standard deviation of 72. The scenarios involve calculating the critical values, test statistics, and drawing conclusions based on the significance thresholds.

a. Critical Values at 1% Significance Level

For a two-tailed test with α = 0.01, the critical z-values are ±2.576, since the standard normal distribution has critical points at these z-scores (Montgomery & Runger, 2018). This means that if the test statistic falls outside the range of -2.576 to +2.576, the null hypothesis will be rejected.

b-1. Calculating the Test Statistic for Sample Mean 294

The test statistic z is calculated as:

z = (x̄ - μ₀) / (σ / √n)

Where:

  • x̄ = 294
  • μ₀ = 260
  • σ = 72
  • n = 85

Calculation:

z = (294 - 260) / (72 / √85) ≈ 34 / (72 / 9.22) ≈ 34 / 7.8 ≈ 4.36

b-2. Conclusion at α=0.01

The calculated z-value of approximately 4.36 exceeds the critical value of 2.576. Since the test statistic is in the rejection region, we reject the null hypothesis at the 1% significance level, indicating that the population mean significantly differs from 260.

c. Critical Values at 10% Significance Level

For α=0.10, the critical z-values are ±1.645. The boundaries for rejection are therefore at these z-scores.

d-1. Calculating the Test Statistic for Sample Mean 238

Using the same formula:

z = (238 - 260) / (72 / √85) ≈ -22 / 7.8 ≈ -2.82

d-2. Conclusion at α=0.10

The calculated z-value of approximately -2.82 falls outside the range of -1.645 to +1.645 (it is less than -1.645). Therefore, we reject the null hypothesis at the 10% significance level, concluding that there is statistically significant evidence that the population mean differs from 260.

Part 2: Comparing the Effectiveness of Two Medications

The second scenario involves a comparison between Relief and SineOut using two independent samples. For each medication, sample sizes, means, and standard deviations are provided. The goal is to determine whether there is a significant difference in the mean time to relief at 5% significance level.

A. Null and Alternative Hypotheses

  • Null hypothesis (H₀): μ₁ = μ₂ (both medications have equal mean relief times)
  • Alternative hypothesis (H₁): μ₁ ≠ μ₂ (the medications differ in mean relief times)

B. Test Statistic

This scenario calls for a two-sample t-test assuming equal variances:

t = (x̄₁ - x̄₂) / √[(s₁² / n₁) + (s₂² / n₂)]

  • x̄₁ = 31.0, s₁ = 5.1, n₁ = 22
  • x̄₂ = 34.3, s₂ = 5.9, n₂ = 22

Calculation:

t = (31.0 - 34.3) / √[(5.1² / 22) + (5.9² / 22)] = (-3.3) / √[(26.01 / 22) + (34.81 / 22)] ≈ -3.3 / √(1.183 + 1.582) ≈ -3.3 / √2.765 ≈ -3.3 / 1.663 ≈ -1.986

C. Decision Rule

The degrees of freedom, using the Welch-Satterthwaite equation, approximate to 42. The critical t-value for a two-tailed test at α=0.05 and df=42 is approximately ±2.018 (Student's t-table, 2019).

If |t| > 2.018, reject H₀; otherwise, fail to reject.

D. Statistical Calculation and Decision

Since |−1.986| ≈ 1.986

E. Conclusions

The analysis suggests that Relief and SineOut do not significantly differ regarding the mean time to relief at the 5% significance level. This indicates comparable effectiveness between the medications within the sample data confines. Further studies with larger samples could provide more definitive insights.

Conclusion

Hypothesis testing serves as a crucial tool in statistical decision-making across various fields. The analyses demonstrated how critical values depend on significance levels, how to compute test statistics, and how to interpret these results in context. The first problem revealed that the population mean significantly differs from 260 under both 1% and 10% significance levels, indicating strong evidence against the null hypothesis. The medication comparison highlighted no statistically significant difference in relief times, emphasizing the importance of empirical data in assessing product efficacy.

References

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