Hypothesis Testing on Unemployment Rate
An analysis was conducted to determine if the unemployment rate among Americans has decreased compared to the previous year. The results indicate a statistically significant decrease in the proportion of unemployed individuals, with a confidence interval supporting the findings. However, considerations such as Type I and Type II errors are discussed in the context of the study, highlighting the complexities of interpreting statistical outcomes in real-world scenarios.
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Presentation Transcript
1) A report on unemployment from last year found that 11% of Americans were out of work at some point in the year. A random sample of 1430 Americans was taken and found that 136 were (a) Does this indicate that Americans are finding more jobs this year? Use a 5% level of significance. H0: p = .11 11% of Americans are unemployed this year. HA: p < .11 Less than 11% of Americans are unemployed this year.
All conditions have been met to use the Normal model for a 1- proportion z-test. Mechanics: 136 1430 0 11 1430 136 0 11 1430 1 800 0 11 0 89 1430 = = = = = p n p . 0 095 . 0 05 . . = = z . ( )( ) . . P-Value = P(z < -1.800) = 0.0359
Conclusion: Since the P-Value is less than alpha(0.0359 < 0.05), we reject the null hypothesis. There is statistically significant evidence that the proportion of Americans that are unemployed is less than 11%. (b) Explain what the P-Value means in this context. There is a 3.6% chance of getting a sample proportion of 9.5% or lower if the true proportion of Americans that are unemployed is truly 11%.
(c) 95% Confidence interval: All conditions have been met to use the Normal model for a 1 proportion z-interval. 136 136 1430 : 1.96 1430 (0.0799,0.1103) 1294 1430 CI 1430 We are 95% confident that the true proportion of Americans that are unemployed lies between 7.99% and 11.03%. (d) Does the conclusion from the confidence interval confirm or contradict your conclusion to the hypothesis test? Explain. It contradicts since 11% falls within the interval.
(e) What should the confidence level have been for the interval to match the significance level for the test? Does the interval created match the conclusion now? Since it was a 1 Sided test and alpha was 5%, should have used a 90% Confidence Level. 136 1430 1294 1430 136 1430 CI : 1.645 1430 (0.0823,0.1079) Now 11% does not fall within the interval. (f) What level of significance would match the 95% confidence level? What would the conclusion to the hypothesis test be if that level of significance was used? = 0.025, Fail to reject
(g) What is a Type I Error in this context? We stated that the proportion of Americans that are unemployed is less than 11% when it is still 11%. (h) What is a Type II Error in this context? We stated that the proportion of Americans that are unemployed is not less than 11% when it is less than 11%. (i) Describe what power means in this context. Power is the probability that we stated that the proportion of Americans that are unemployed is less than 11% when it is less than 11%.
(j) Suppose that a new sample was taken with 621 Americans. A new test is run with the same level of significance(5%). How would the Type I Error, Type II Error, and Power be affected by this change? Type I Error Remains the same Type II Error Increases Power Decreases
