Stat 200 Week 5 Homework Problems According To February

Stat 200 Week 5 Homework Problems712according To The February 2008 F

Analyze statistical data and hypotheses related to consumer fraud, identity theft, Autism Spectrum Disorder incidence, economic dynamism, and balance stability among elderly individuals. Conduct hypothesis tests at specified significance levels, interpret confidence intervals, and evaluate the effects of changing sample sizes on interval estimates, applying statistical reasoning to real-world datasets.

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The statistical problems provided encompass various real-world scenarios involving proportions, means, and hypothesis testing, illustrating the application of inferential statistics in social, health, and economic contexts. Each problem requires formulating hypotheses, conducting significance tests, interpreting confidence intervals, and understanding the implications of statistical decisions and parameter changes.

Firstly, examining the claims related to consumer fraud and identity theft in Alaska involves hypothesis testing about population proportions. The data indicate that in 2007, Alaska had a proportion of identity theft complaints approximately 22.4% (321 complaints out of 1,432), which is slightly below the national figure of 23%. To analyze whether this difference is statistically significant, a one-proportion z-test is appropriate. The null hypothesis (H0) assumes that Alaska's proportion equals the national average (p = 0.23), while the alternative hypothesis (Ha) posits that the proportion is less than 0.23, indicating a lower rate in Alaska. The significance level (α) typically used is 0.05, which balances the risk of Type I error—incorrectly rejecting a true null hypothesis—against statistical confidence.

Performing the hypothesis test involves calculating the test statistic using the sample proportion, the standard error, and comparing it to the critical value or p-value. Given the sample size, the z-test assesses whether the observed difference could have arisen by chance. If the p-value is less than 0.05, the evidence suggests Alaska's proportion of identity theft complaints is statistically significantly lower than the national average.

Secondly, understanding the risks associated with Type I and Type II errors in this context is essential. A Type I error (false positive) would mean concluding Alaska's rate is lower when it is not, potentially leading to complacency or inadequate policy responses. A Type II error (false negative) would mean failing to detect a true difference, possibly resulting in overlooked opportunities for intervention. The consequences of these errors emphasize the importance of selecting an appropriate α, such as 0.05, to control the likelihood of Type I errors, especially when policy implications are significant.

Next, the analysis of ASD prevalence in Arizona compared to national levels involves calculating confidence intervals for proportions. With 507 cases out of 32,601 children, the sample proportion is approximately 1.55%. A 99% confidence interval provides a range within which the true proportion is estimated to lie with high certainty. The interval is constructed using the sample proportion, the standard error, and the z-score corresponding to 99% confidence (approximately 2.576). If the lower bound of this interval exceeds the national incidence rate of 1 in 88 (about 1.14%), it indicates strong evidence that ASD prevalence in Arizona is higher than the national average, with the statistical significance confirmed at the 1% level.

The economic dynamism of middle-income countries compares to high-income counterparts involves testing whether the mean economic growth index is lower in middle-income nations. A one-sample t-test compares the sample mean from the data to the known population mean of high-income countries (60.29). The null hypothesis states that the mean dynamism is equal or greater than 60.29, while the alternative suggests it is less. Conducting this test at the 5% significance level involves calculating the t-statistic based on the sample mean, standard deviation, and sample size, and then comparing it to the critical t-value or p-value.

Similarly, for elderly balance stability, the mean sway in millimeters is tested against a known mean (18.125 mm). A hypothesis test assesses whether the elderly sway more, with a 5% significance level. The calculation involves deriving the t-statistic from the sample mean, standard deviation, and sample size, then making a decision based on the critical value.

Regarding the impact of sample size on confidence intervals, decreasing the sample size from 100 to 80 affects the margin of error, which is proportional to the standard error (dependent on the square root of the sample size). A smaller sample size increases the standard error, widening the confidence interval, thus reducing the precision of the estimate.

Interpreting confidence intervals in the context of public opinion, such as Gallup's survey on the government's responsibility for healthcare, involves understanding that the interval provides a range of plausible values for the true population proportion, given a certain confidence level (e.g., 95%). This means we are confident that the true proportion of Americans believing healthcare responsibility lies with the government falls within this range, based on the sample data.

Finally, in estimating the proportion of children diagnosed with ASD in Arizona with a 99% confidence level, the calculation employs the sample proportion and standard error, incorporating the z-value for 99% confidence (2.576). The resulting confidence interval offers a precise estimate of the ASD prevalence, which can guide healthcare policy and resource allocation.

Overall, these statistical analyses demonstrate the importance of hypothesis testing and confidence intervals in making data-driven decisions and understanding the significance of differences and estimates in diverse fields, including public health, economics, and social sciences.

References

  • Centers for Disease Control and Prevention (CDC). (2013). Features - Autism Spectrum Disorder. https://www.cdc.gov/ncbddd/autism/facts.html
  • Consumer Fraud and Identity Theft. (2008). Federal Trade Commission Report, February 2008.
  • Maintaining Balance While Aging Study. (2013). SOCR Data Repository.
  • Autism and Developmental Disorders. (2008). State of Arizona Health Department.
  • Socioeconomic Data and Research. (2013). World Bank. Economic Dynamism Data.
  • Gallup. (2013). Public Opinion on Government Responsibility for Healthcare. Gallup Poll.
  • National Autism Association. (2013). Autism Prevalence Estimates.
  • Sample Size Influence on Confidence Intervals. (2013). SOCR Data.
  • Hypothesis Testing Principles. (2010). Statistical Methods in Public Health.
  • Statistical Inference and Confidence Intervals. (2012). Authoritative Textbook on Statistics.

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