Paper For Above instruction
The aim of this paper is to analyze two statistical scenarios based on hypothesis testing to determine whether certain claims can be supported by data. The first scenario examines whether a new drug effectively lowers body temperature in individuals suffering from a cold virus, while the second assesses whether more than half of a population supports a specific land-use decision.
Scenario 1: Effectiveness of a New Cold Drug in Lowering Body Temperature
The null hypothesis (H■) posits that the drug has no effect, meaning the population mean body temperature remains at 99.5°F. Mathematically, H■: µ = 99.5. The alternative hypothesis (H■) asserts that the drug lowers body temperature, such that µ < 99.5°F.
Given the data: sample size (n) = 25, sample mean (x■) = 98.5°F, sample standard deviation (s) = 1.0°F, and significance level (α) = 0.01, we perform a one-sample t-test to evaluate the hypotheses. The t-statistic is calculated as:
t = (x■ - µ■) / (s / √n) = (98.5 - 99.5) / (1.0 / √25) = (-1.0) / (0.2) = -5.0
Using a t-distribution table or statistical software, the p-value corresponds to the probability of observing a t-value as extreme as -5.0 under the null hypothesis. For degrees of freedom (df) = 24, this p-value is very small, approximately less than 0.001.
Since p < α = 0.01, we reject the null hypothesis, providing strong evidence that the drug is effective in lowering body temperature in cold sufferers.
Scenario 2: Support for Land-Use Decision
The null hypothesis (H■) states that half or fewer people support the decision, H■: p ≤ 0.5. The
alternative hypothesis (H■) suggests that more than half support it, H■: p > 0.5.
From the sample, 55 out of 80 individuals support the decision, resulting in a sample proportion: p■ = 55 / 80 = 0.6875
The test statistic for proportions is: z = (p■ - p■) / √(p■(1 - p■) / n) = (0.6875 - 0.5) / √(0.5 * 0.5 / 80) ≈ 0.1875 / 0.0559 ≈ 3.355
Corresponding p-value for a one-tailed test (z > 3.355) is approximately 0.0004.
Since p < α = 0.05, we reject H■, concluding that there is sufficient evidence that more than half of the population supports the land-use decision.
Conclusion
Based on the hypothesis tests, the drug appears effective in reducing body temperature among cold sufferers, with strong statistical significance. Simultaneously, there is compelling evidence to support that a majority of the population favors the land-use decision. These conclusions are made with confidence levels set at 1% and 5% respectively, adhering to rigorous statistical standards.
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