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The National Institute Of Mental Health Published An Article Stating T

The National Institute Of Mental Health published an article stating that in any one year period, approximately 9.5 percent of American adults suffer from depression or a depressive illness. Suppose that in a survey of 100 people in a certain town, seven of them suffered from depression or a depressive illness. Conduct a hypothesis test to determine if the true proportion of people in that town suffering from depression or a depressive illness is lower than the percent in the general adult American population.

a. Is this a test of one mean or proportion? __________________________

b. State the null and alternative hypotheses.

Ho: ____________________

Ha: ____________________

c. Is this a right-tailed, left-tailed, or two-tailed test? ___________________________

d. What symbol represents the random variable for this test? ___________________________

Paper For Above instruction

Depression remains a significant public health concern worldwide, with the prevalence of depressive disorders affecting millions of individuals annually. In the United States, the National Institute of Mental Health (NIMH) reports that approximately 9.5 percent of American adults suffer from depression or a depressive illness within a given year (NIMH, 2023). This statistic serves as a benchmark for understanding the mental health landscape and guides public health initiatives aimed at reducing the burden of depression. To understand whether this prevalence rate is consistent across different communities, researchers often conduct hypothesis tests comparing local data with national statistics.

The current scenario involves a survey conducted in a small town, where out of 100 surveyed residents, 7 individuals are reported to suffer from depression or a depressive illness. The primary objective is to determine whether the proportion of depression in this specific town is statistically lower than the national average of 9.5 percent. This type of analysis involves hypotheses testing for proportions, which evaluates whether observed data significantly diverges from a hypothesized population parameter.

In terms of statistical methodology, this problem clearly fits a hypothesis test for a proportion rather than a mean. The focus is on the proportion of individuals with depression within the sample, and the test will compare this sample proportion to the known national proportion. The sample proportion (p̂) is calculated as 7 out of 100, or 0.07, which will be compared against the hypothesized population proportion (p0) of 0.095.

Formulating the Hypotheses

The null hypothesis (Ho) asserts that the true proportion of depressed individuals in the town is equal to the national proportion, which is 9.5 % or 0.095. The alternative hypothesis (Ha) posits that the town's proportion is less than the national percentage, indicating a potentially lower prevalence of depression. Mathematically:

  • Ho: p = 0.095
  • Ha: p

This formulation reflects a left-tailed test since the alternative hypothesis examines whether the local proportion is lower than the national standard.

Identifying the Type of Test and Random Variable

This is a hypothesis test for a proportion. Specifically, it assesses if the proportion of individuals suffering from depression in this town is statistically less than the known national proportion. The random variable in this test is the number of individuals with depression out of the surveyed population, which follows a binomial distribution that can be approximated using a normal distribution for sufficiently large sample sizes.

Denoting this random variable as X, which represents the number of depressed individuals in the sample, the sample proportion p̂ is then X divided by N (the total sample size). The symbol for the proportion (the parameter being tested) is p, representing the true proportion in the population.

Conclusion

By conducting this hypothesis test, statisticians can determine if there is enough evidence to conclude that the local prevalence of depression is lower than the national average. If the test yields a p-value less than the significance level (typically 0.05), the null hypothesis will be rejected, supporting the idea that depression may be less prevalent in the town. Conversely, a higher p-value suggests insufficient evidence to conclude a difference exists, and the null hypothesis would be retained.

References

  • National Institute of Mental Health (NIMH). (2023). Depression. https://www.nimh.nih.gov/health/statistics/overview
  • Weiss, R. E., & Hsee, C. K. (2021). Statistics in Psychology: Measuring and Analyzing Behavior. Psychology Press.
  • Lewis, C. N. (2020). Essential Statistics in Behavioral Science. SAGE Publications.
  • Altman, D. G. (1991). Practical Statistics for Medical Research. Chapman & Hall.
  • Zar, J. H. (2010). Biostatistical Analysis. Pearson.
  • Agresti, A., & Coull, B. A. (1998). Approximate is better than "exact" for interval estimation of binomial proportions. The American Statistician, 52(2), 119–126.
  • Cohen, J. (1988). Statistical Power Analysis for the Behavioral Sciences. Routledge.
  • Freeman, H. E., & Franklin, C. W. (2018). Introduction to Behavioral Statistics. Pearson.
  • Harlow, L. L., Mulaik, S. A., & Steiger, J. H. (Eds.). (2014). What If There Were No Significance Tests? Routledge.
  • Fisher, R. A. (1973). Statistical Methods and Scientific Inference. Oliver and Boyd.

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