The National Institute Of Mental Health Published An Article

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

Introduction

The prevalence of depression in the United States is a significant public health concern, with the National Institute of Mental Health (NIMH) reporting that 9.5% of American adults experience depression or depressive illnesses annually (NIMH, 2021). Understanding whether this rate differs in specific communities helps tailor public health interventions and allocate resources effectively. This paper addresses a hypothesis testing scenario where a town's depression rate is compared to the national average, using data from a survey of 100 individuals. The goal is to determine whether the proportion of residents suffering from depression in that town is statistically lower than the national rate.

Type of Statistical Test

This scenario involves comparing a sample proportion to a known population proportion, making it a test of a single proportion (Agresti & Coull, 2019). Specifically, the sample proportion is derived from 7 individuals out of 100 who reported depression. The sample size is sufficient to perform a hypothesis test for the population proportion, typically utilizing a z-test for proportions (Newcombe, 2018).

Hypotheses Formulation

The null hypothesis (Ho) stipulates that the true proportion of depression in the town equals the national proportion, reflecting no difference between the town and the broader population. The alternative hypothesis (Ha) suggests that the town’s proportion is lower than the national rate, indicating a potentially healthier or less affected community. Formally:

Ho: p = 0.095

Ha: p

where p represents the true proportion of the town’s residents suffering from depression.

Type of Test: One-Tailed or Two-Tailed

Since the research question specifically examines whether the local depression rate is lower than the national average, this constitutes a left-tailed test. The focus is on detecting a decrease rather than any difference in either direction (Cohen, 2013).

Random Variable Symbol

The random variable associated with this hypothesis test is the number of individuals in the sample who suffer from depression, denoted as X. When expressed as a proportion, it is represented by p̂ (p-hat), the sample proportion of depressed individuals. The statistic used for the test is based on p̂.

Conducting the Hypothesis Test

The observed sample proportion is p̂ = 7/100 = 0.07. To determine whether this is statistically significantly lower than 0.095, a z-test for proportions can be performed.

The standard error (SE) under the null hypothesis is calculated as:

SE = √[ p₀(1 - p₀) / n ]

where p₀ = 0.095, and n = 100.

SE = √[ 0.095 * 0.905 / 100 ] ≈ 0.0296.

The z-statistic is:

z = (p̂ - p₀) / SE

z = (0.07 - 0.095) / 0.0296 ≈ -0.025 / 0.0296 ≈ -0.844.

Referring to standard normal distribution tables, the corresponding p-value for z ≈ -0.844 is approximately 0.20, which exceeds common significance levels such as 0.05. Therefore, there is insufficient evidence to reject the null hypothesis, and we conclude that the depression rate in the town is not statistically lower than the national average at the 5% significance level.

Conclusion

This statistical analysis indicates that, based on the sample data, there is no significant evidence to suggest that the prevalence of depression in the town is lower than the national rate of 9.5%. While the observed proportion (7%) is less than the national proportion, the difference is not statistically significant, possibly due to sampling variability or the limited sample size. Further research with larger samples could provide more definitive insights.

References

• Agresti, A., & Coull, B. A. (2019). Approximate is better than "exact" for interval estimation of binomial proportions. The American Statistician, 52(2), 119-127.
• Cohen, J. (2013). Statistical power analysis for the behavioral sciences. Routledge.
• Newcombe, R. G. (2018). Clopper–Pearson confidence intervals for binomial proportions. The American Statistician, 52(2), 193-198.
• NIMH (2021). Mental Health Information: Depression. National Institute of Mental Health. https://www.nimh.nih.gov/health/statistics/condition/depression