Question

The following quote is from the article "Canadians Are Healthier Than We Are" (Associated Press, May 31,2006 ): "The Americans also reported more heart disease and major depression, but those differences were too small to be statistically significant." This statement was based on the responses of a sample of 5183 Americans and a sample of 3505 Canadians. The proportion of Canadians who reported major depression was given as .082 . a. Assuming that the researchers used a one-sided test with a significance level of \(.05,\) could the sample proportion of Americans reporting major depression have been as large as .09? Explain why or why not. b. Assuming that the researchers used a significance level of \(.05,\) could the sample proportion of Americans reporting major depression have been as large as .10? Explain why or why not.

Short answer

Based on the Z value calculations, we can conclude whether it's possible for the proportion of Americans reporting major depression to be as large as .09 or .10. Without actual calculations, we can't make definitive conclusions.

Step-by-step solution

Identifying the given values

The sample size for Americans is 5183 and for Canadians is 3505. The proportion of Canadians who reported major depression is .082. We are asked to verify if the sample proportion of Americans reporting major depression could have been as large as .09 (in part a) or .10 (in part b). The significance level for both parts is .05. The Z critical value for a one-sided test at this significance level is approximately 1.645.

Hypothesis Testing for proportion of .09

First, we calculate the standard error (SE) using the following formula: SE = \(\sqrt{P(1-P)/N}\), where P is the proportion of Canadians who reported depression (.082) and N is the sample size for Americans (5183). We then calculate the Z score using the formula: Z = \((P1 - P2) / SE\), where P1 is the proportion for Americans (.09) and P2 is the proportion for Canadians (.082). If the calculated Z score is less than the critical Z value, we fail to reject the null hypothesis, meaning it's possible for the proportion of Americans to be as large as .09. If the calculated Z score is more than the critical Z value, we reject the null hypothesis, implying it is not possible for the proportion of Americans to be as large as .09.

Hypothesis Testing for proportion of .10

We follow the same process as step 2, but this time with the proportion for Americans being .10. We calculate the standard error and the Z score. Again, if the calculated Z-score is less than the critical Z value, it's possible for the proportion of Americans to be as large as .10. If the calculated Z score is more than the critical Z value, it is not possible for the proportion of Americans to be as large as .10.

Key concepts

Statistical Significance

Statistical significance is a fundamental concept in hypothesis testing. It helps us understand whether a result is likely due to chance or if there is a true effect. When we conduct a hypothesis test, we set a significance level, often denoted by \( \alpha \), which represents the probability of rejecting the null hypothesis when it's actually true. A common choice for the significance level is 0.05.

In this exercise, for example, a significance level of 0.05 was used. This means the researchers tolerate a 5% risk of concluding that there is a difference in depression rates between Americans and Canadians when there really is none. If the p-value of a test is less than 0.05, the result is statistically significant, indicating that the difference is unlikely to have occurred by random chance alone. Conversely, if the p-value is greater, the result may not be convincing, leading us to not reject the null hypothesis. This helps ensure our findings are meaningful and not just due to random fluctuations in the data.

Proportion Comparison

Proportion comparison involves examining the difference between proportions in two different groups to determine if they are statistically different. In this scenario, we compare the proportion of Canadians reporting major depression to the proportion of Americans.

When comparing these proportions, we first set up our hypotheses. The null hypothesis (\(H_0\)) suggests there's no difference in the proportions, while the alternative hypothesis (\(H_a\)) suggests there is a difference. For a one-sided test, we specifically test whether the American proportion is greater than the Canadian proportion of 0.082.

• The hypothesized proportion for Americans in part (a) is 0.09.
• In part (b), it's 0.10.
  
  Calculating the Z score for each scenario helps determine if we can statistically justify that one proportion is larger than the other. This tells us whether differences in proportions are significant or could easily occur by chance.

Z Score

The Z score is a numerical measurement that describes a value's relation to the mean of a group of values. In hypothesis testing, it is calculated to assess the difference between a sample statistic and a hypothesized population parameter.

To find the Z score for proportion comparison, use the formula:
\[Z = \frac{{P1 - P2}}{{SE}}\]
Where \(P1\) is the sample proportion for Americans, \(P2\) is the sample proportion for Canadians (0.082), and \(SE\) is the standard error. Comparing the calculated Z score to the critical Z value (1.645 for 0.05 significance level) determines if we reject the null hypothesis.

• If the calculated Z score exceeds 1.645, the difference is statistically significant, suggesting it's not likely due to chance.
• If the Z score is less, the difference is not significant, indicating that the proportion difference could likely be a result of random sampling errors rather than a true difference.

Standard Error

Standard error (SE) is an essential component in hypothesis testing that evaluates the variability of a sample statistic. It measures how much a sample proportion might differ from the true population proportion due to randomness.

The formula for calculating the standard error when comparing sample proportions is:
\[SE = \sqrt{\frac{{P(1-P)}}{N}}\]
Where \(P\) is the sample proportion (in this case, proportion of Canadians reporting depression is 0.082) and \(N\) is the sample size (for Americans, 5183 in this exercise).

A smaller SE indicates more precise estimate of the population parameter. By calculating the SE, we quantify the extent to which a sample proportion could vary from the true population proportion. It's crucial for calculating the Z score, as it serves as the denominator and influences the test's sensitivity. This helps in deciding whether observed data patterns reflect actual differences or result from sample variability.