Financial Reporting & Analysis Using Financial Accounting

Financial Reporting & Analysis Using Financial Accounting Information 13th Edition

In 2004, a random sample of 405 community residents was obtained, with 275 expressing confidence in being notified quickly of a radioactive incident. In 2005, after the emergency preparedness program started, a sample of 388 residents was taken, with 310 feeling confident they would be notified promptly. The goal is to determine, at a 5% significance level, whether there is evidence that the proportion of residents confident in 2005 is greater than that in 2004.

Paper For Above instruction

The problem presented involves comparing two population proportions to assess whether there is an increase in residents' confidence levels following the implementation of a community emergency program. Specifically, the question asks if the proportion of residents confident about being notified quickly in 2005 is significantly greater than in 2004, using a 5% significance level. This type of problem corresponds to a hypothesis test for the difference between two proportions, which is a common statistical method used to compare two independent groups' proportions in inferential statistics.

Part a: Type of Statistical Test and Key Indicators

This problem is a hypothesis test for the difference between two population proportions. The key words that identify this are "is there evidence that the true proportion in 2005 is greater than in 2004" and "at the 5% significance level." The phrase "greater than" indicates a one-tailed test, particularly an upper-tail test, since the alternative hypothesis posits that the proportion in 2005 exceeds that in 2004. The focus is on determining if the observed difference in sample proportions is statistically significant enough to infer a true increase in the population proportion.

Part b: Procedure Name and Parameters

The appropriate procedure here is a two-proportion z-test. This test compares the sample proportions from two independent samples to assess whether the difference between these proportions is statistically significant. The parameters involved include:

• Sample sizes: n₁ = 405 (2004), n₂ = 388 (2005)
• Sample proportions: p̂₁ = 275/405 ≈ 0.679, p̂₂ = 310/388 ≈ 0.799

The procedure involves calculating a pooled proportion under the null hypothesis (that the population proportions are equal), then computing the test statistic (z-value) based on the difference in sample proportions, adjusted by the standard error of the difference.

Part c: Hypothesis and Test Statistic Details

Since the research question explores if the proportion in 2005 is greater than in 2004, the hypotheses are set as:

• Null hypothesis (H₀): p₂ - p₁ = 0 (no difference)
• Alternative hypothesis (H₁): p₂ - p₁ > 0 (proportion in 2005 is greater)

This constitutes a right-tailed test. The test statistic formula for the z-test for two proportions is:

z = (p̂₂ - p̂₁) / SE

where the standard error (SE) is calculated as:

SE = √[P̂_pooled (1 - P̂_pooled) (1/n₁ + 1/n₂)]

and the pooled proportion P̂_pooled is given by:

P̂_pooled = (x₁ + x₂) / (n₁ + n₂) = (275 + 310) / (405 + 388) ≈ 585 / 793 ≈ 0.738

Conclusion

In summary, this problem is a right-tailed hypothesis test for the difference between two population proportions, using a two-proportion z-test. The key steps involve calculating the sample proportions, pooled proportion, the test statistic, and comparing it to the critical value at the 5% significance level to determine whether to reject the null hypothesis. Given the sample proportions, the observed difference suggests an increase in confidence in 2005, but formal statistical testing would confirm whether this increase is statistically significant.

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