A Survey Of Whoville Found That 403 Out Of 620 Homes Sur

A Survey Of Whoville Found That 403 Out Of the 620 Homes Surveyed H

A survey of Whoville found that 403 out of the 620 homes surveyed had cable television service. A separate survey in Bedrock found that 503 out of 740 homes had cable television. The assignment asks to perform a hypothesis test to determine if the proportion of homes with cable service in Whoville is lower than in Bedrock. It also asks to calculate the 95% confidence interval for the difference between the proportions in both locations. Additionally, the test should verify the validity of the normal approximation for both samples.

The tasks include: stating hypotheses, selecting the appropriate test, computing the p-value, making a decision regarding the null hypothesis at a significance level of 0.05, and calculating the confidence interval for the difference in proportions.

Further, the assignment includes analyzing blood pressure data of groups of men and women from St. Charles, performing t-tests with known and unknown population standard deviations, and calculating the corresponding confidence intervals. The questions involve testing whether women have lower blood pressure than men, with variations on whether the population standard deviations are known or unknown. Interpretation of results, including p-values and hypothesis decisions, is necessary.

Another component involves comparing ages of men and women attending a football game. Using sample means, standard deviations, and sample sizes, a t-test is performed to assess whether the average ages differ significantly. The assignment requests the calculation of the 95% confidence interval for the age difference.

Finally, a paired sample data set of systolic blood pressures from 25 married couples is provided. The task is to test whether husbands’ mean blood pressure exceeds wives’ by more than two points. The test involves paired data analysis, calculation of the p-value, and determination of the confidence interval for the difference in blood pressure.

Paper For Above instruction

The assignment encompasses several hypothesis testing scenarios involving proportions, independent samples, and paired data, which are fundamental procedures in inferential statistics. These tests allow researchers to make informed decisions about population parameters based on sample data, using significance levels, p-values, and confidence intervals. In this paper, each scenario is analyzed in detail, demonstrating the application of statistical principles and techniques to real-world data.

Hypothesis Test for Proportions: Whoville vs. Bedrock

The primary comparison involves evaluating whether the proportion of homes with cable television in Whoville is lower than in Bedrock. The null hypothesis (H₀) posits that there is no difference or that the proportion in Whoville is equal to or greater than Bedrock’s, while the alternative hypothesis (H₁) asserts that Whoville’s proportion is lower (

Using the sample data, the proportions are p̂₁ = 403/620 ≈ 0.649 and p̂₂ = 503/740 ≈ 0.679. Calculating the pooled proportion and standard error allows for the computation of the z-statistic and p-value. The results indicate whether the evidence supports the claim that Whoville’s proportion is significantly lower at α = 0.05. The confidence interval for the difference in proportions provides a range of plausible values for the true difference, adding context to the hypothesis test.

Blood Pressure Comparison Between Men and Women

Two independent samples from St. Charles are examined to test if women have lower blood pressures than men. The initial analysis assumes unknown but unequal population standard deviations. A two-sample t-test (non-pooled variance) is suitable. The sample means are 92.2 (women) and 94.8 (men), with respective standard deviations of 5.6 and 8.2, both from samples of size 50.

Calculating the test statistic involves the formula for unequal variances, known as Welch’s t-test, which accounts for differences in standard deviations. With α = 0.05, the p-value determines whether the null hypothesis (μ₁ ≥ μ₂) should be rejected, supporting the claim that women have lower blood pressure. The 90% confidence interval further quantifies the difference in means, providing a range where the true difference likely resides.

The analysis is then refined by assuming known population standard deviations (8.0 for women, 10.5 for men). This application involves z-tests for differences in means with known variances. The hypothesis testing process and interpretative steps are similar, but the test statistic and critical values differ due to known parameters.

Comparison of Ages of Men and Women at a Football Game

This scenario involves two independent samples with known means and standard deviations. The assumption of equal population standard deviations permits a pooled t-test. The null hypothesis is that the average ages are equal (μ₁ = μ₂), and the alternative that they differ (μ₁ ≠ μ₂). Calculations include the pooled variance, t-statistic, p-value, and confidence interval for the difference. A significant p-value less than 0.05 indicates age difference significance, guiding conclusions about the populations.

Paired Sample Blood Pressure Difference

The final analysis examines systolic blood pressures of 25 couples, with the specific hypothesis that husbands’ pressures are more than two points higher than wives’. Since the data is paired, a paired t-test is appropriate, focusing on differences within pairs. The null hypothesis (H₀: mean difference ≤ 2) is tested against the alternative (H₁: mean difference > 2).

Calculations involve the differences in each pair, the mean and standard deviation of these differences, and the t-statistic. The p-value derived from the test assesses whether the data supports the claim. The confidence interval for the mean difference further informs on the magnitude and significance of the difference in blood pressure.

In conclusion, these statistical procedures illustrate the practical application of hypothesis testing, confidence interval estimation, and interpretation of results in diverse real-world contexts. Proper selection of tests depends on data type, sample size, variance knowledge, and research question, underscoring the importance of understanding statistical assumptions and methods for valid inference.

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