A Researcher Claims That 62% Of Voters

A Researcher Claims That 62 Of Voters

A researcher claims that 62% of voters favor gun control. State the null hypothesis and the alternative hypothesis in words.

Identify the null hypothesis H₀ and the alternative hypothesis H₁ using p.

The city claims that its new public transportation system will reduce traffic congestion. An analysis shows a 0.04 p-value. Determine whether the results are statistically significant at the 0.05 significance level.

A survey indicates that 35% of adults in a country prefer online shopping. A researcher tests this claim with a sample of 200 adults, finding that 40% prefer online shopping.

Sample data: n = 200, p̂ = 0.40, claimed proportion p₀ = 0.35.

Calculate the test statistic for this hypothesis test.

Using a significance level of 0.05, find the p-value, determine whether to reject the null hypothesis, and interpret the result in a non-technical manner.

Paper For Above instruction

Introduction

Hypothesis testing is a fundamental aspect of statistical analysis, used to make inferential decisions about population parameters based on sample data. It involves formulating null and alternative hypotheses, calculating a test statistic, comparing it with critical values or p-values, and finally making a decision regarding the validity of the claim. This paper demonstrates these concepts using a variety of real-world examples, including political preferences, transportation benefits, and consumer behavior.

Formulating Hypotheses

The initial step in hypothesis testing is to set the null hypothesis (H₀) and the alternative hypothesis (H₁). The null hypothesis typically asserts that there is no effect or no difference, acting as a statement of status quo or no association. Conversely, the alternative hypothesis represents the claim or suspicion being tested, indicating a potential effect or difference.

For instance, if a researcher claims that 62% of voters favor gun control, the hypotheses would be:

- H₀: p = 0.62 (the proportion of voters favoring gun control is 62%)

- H₁: p ≠ 0.62 (the proportion differs from 62%)

Similarly, in the case of Carter Motor Company's claim regarding the new sedan's fuel efficiency, the hypotheses might be:

- H₀: μ ≤ 30 miles per gallon

- H₁: μ > 30 miles per gallon

The selection of one- or two-tailed tests depends on the nature of the claim.

Evaluating Significance and Errors

A crucial aspect of hypothesis testing involves determining statistical significance, often using a p-value, which indicates the probability of observing the sample data assuming the null hypothesis is true. A small p-value (less than the chosen significance level, α) suggests that the observed data are unlikely under the null hypothesis, leading to rejection of H₀.

For example, a city claims that its new public transportation system reduces traffic congestion, and an analysis yields a p-value of 0.04. At α = 0.05, since 0.04

A Type I error occurs if the null hypothesis is rejected when it is actually true, while a Type II error occurs if it is not rejected when it is false. Understanding these errors helps in interpreting the results of hypothesis tests accurately and cautiously.

Hypothesis Testing for Population Proportions

In testing proportions, the z-test for one proportion is typically used when the sample size is large, and the sampling distribution approximates normality. The test statistic is calculated as:

z = (p̂ - p₀) / √[p₀ (1 - p₀) / n]

where p̂ is the sample proportion, p₀ is the claimed proportion, and n is the sample size.

In the example where 40% of 200 adults favor online shopping, with a claimed proportion of 35%, the test statistic is calculated as follows:

z = (0.40 - 0.35) / √[0.35 * 0.65 / 200] ≈ 1.91

Using the standard normal distribution, the p-value associated with this z-value indicates whether to reject H₀ at the chosen significance level.

Decision Making and Interpretation

After calculating the test statistic and p-value, compare the p-value to α:

- If p-value 0: there is sufficient evidence to support the alternative hypothesis.

- If p-value ≥ α, fail to reject H₀: there is insufficient evidence to support the alternative.

In the online shopping example, with a p-value corresponding to z = 1.91, the p-value might be approximately 0.056. Since 0.056 > 0.05, we fail to reject H₀, indicating that the data do not provide strong enough evidence to conclude that the proportion who prefer online shopping exceeds 35%.

Interpreting these results in real-world terms involves understanding the limitations, such as the probability of Type I and Type II errors, and the context of the data.

Conclusion

Hypothesis testing provides a structured framework for making data-driven decisions about population parameters. Proper formulation of hypotheses, calculation of test statistics, interpretation of p-values, and understanding potential errors are all vital components of valid statistical inference. Applying these methods ensures that conclusions drawn are supported by evidence, ensuring validity and reliability in research.

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