Researchers Interviewed Young Adults In Canada And The US

Researchers Interviewed Young Adults In Canada And The Us To Compare

Researchers interviewed young adults in Canada and the U.S. to compare their ages when they enter the workforce. The mean age of the 100 Canadians was 18 with a standard deviation of 6. The mean age of the 130 people interviewed in the U.S. was 20 with a standard deviation of 8. Is the mean age of entering the workforce in Canada lower than the mean age in the U.S.?

a) Identify the null and alternate hypothesis

b) Find the test statistic (show all work)

c) Find the P-value (show all work) and evaluate whether the mean age of entering the workforce in Canada is lower than the mean age in the U.S.? Test at a 1% significance level.

Paper For Above instruction

The research question seeks to determine whether young adults in Canada enter the workforce at a younger age than their counterparts in the United States. To address this, a hypothesis test comparing the means of two independent samples is appropriate. The analysis involves several steps: formulating hypotheses, computing the test statistic, identifying the p-value, and making a conclusion based on the significance level.

Formulation of Hypotheses

The null hypothesis (H₀) assumes there is no difference in the mean ages at which young adults in Canada and the U.S. enter the workforce:

H₀: μ₁ = μ₂

The alternative hypothesis (H₁) posits that Canadian young adults enter the workforce at an earlier age:

H₁: μ₁

Here, μ₁ represents the population mean age in Canada, and μ₂ represents the population mean age in the U.S. This is a one-tailed test because the research question emphasizes whether the Canadian mean is lower.

Data Summary

- Canada: n₁=100, mean₁=18, SD₁=6

- U.S.: n₂=130, mean₂=20, SD₂=8

Calculation of the Test Statistic

Given the data, the test involves the comparison of two independent samples with known sample sizes and standard deviations. Since the standard deviations are known, or more appropriately, the sample sizes are large enough, a Z-test can be used.

The formula for the Z-test statistic for two independent means is:

Z = (mean₁ - mean₂) / √( (SD₁² / n₁) + (SD₂² / n₂) )

Calculating the numerator:

mean₁ - mean₂ = 18 - 20 = -2

Calculating the denominator:

SD₁² / n₁ = (6)² / 100 = 36 / 100 = 0.36

SD₂² / n₂ = (8)² / 130 = 64 / 130 ≈ 0.4923

Sum:

0.36 + 0.4923 ≈ 0.8523

Standard error:

√0.8523 ≈ 0.9232

Calculating the Z-value:

Z = -2 / 0.9232 ≈ -2.166

This Z-value indicates how many standard errors the sample difference is below zero.

Calculating the P-value

Using standard normal distribution tables or a calculator, the p-value associated with Z ≈ -2.166 for a left-tailed test is approximately 0.0152.

This p-value indicates the probability of observing a difference as extreme as -2 or more in favor of the alternative hypothesis if the null hypothesis is true.

Conclusion at 1% Significance Level

Since the p-value (≈ 0.0152) exceeds the significance level of 0.01, we do not reject the null hypothesis. There is insufficient evidence at the 1% level to conclude that Canadian young adults enter the workforce at a lower age than U.S. young adults.

Final Remarks

While the data suggests that Canadians tend to enter the workforce at a younger age on average, the statistical evidence does not support this conclusion with high confidence at the 1% significance level. Future studies could incorporate larger samples or refine the measurement of 'age when entering the workforce' to obtain more conclusive results.

References

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