A federal funding program is available to low-income...

A federal funding program is available to low-income neighborhoods. To qualify for the funding, a neighborhood must have a mean household income of less than $$\$ 15,000$$ per year. Neighborhoods with mean annual household income of $$\$ 15,000$$ or more do not qualify. Funding decisions are based on a sample of residents in the neighborhood. A hypochesis test with a .02 level of significance is conducted. If the funding guidelines call for a maximum probability of .05 of not funding a neighborhood with a mean annual housebold income of $$\$ 14,000$$, what sample size should be used in the funding decision study? Use $$\sigma-\$ 4000$$ as a planning value.

Key Concepts

Sample Size Determination

Sample size determination is the process of calculating the minimum number of observations required in a study to achieve a desired level of power or precision. In the context of hypothesis testing and power analysis, it involves balancing the probabilities of Type I and Type II errors while considering the expected effect size and variability in the data. Proper sample size is crucial to ensure that a study is efficient and capable of reliably detecting the effect of interest.

Power Analysis

Power analysis is a technique used to determine the probability that a test will detect an effect or difference when one truly exists. It involves calculating the power, which is one minus the probability of a Type II error, given a specific effect size, significance level, and sample size. This analysis is essential for planning studies to ensure that they are capable of detecting meaningful differences or effects in the population.

Type I and Type II Errors

Type I and Type II errors are two types of errors that can occur in hypothesis testing. A Type I error happens when the null hypothesis is incorrectly rejected when it is actually true, while a Type II error occurs when the null hypothesis is not rejected even though the alternative hypothesis is true. Minimizing these errors is critical in decision making; the significance level (alpha) controls the probability of a Type I error, and power analysis relates to controlling the probability of a Type II error.

Hypothesis Testing

Hypothesis testing is a statistical method used to make decisions or inferences about a population based on sample data. It involves formulating a null hypothesis (which represents a baseline or default position) and an alternative hypothesis, and then determining whether the observed sample data provide sufficient evidence to reject the null hypothesis. This framework is essential for assessing claims and making decisions in various fields by controlling error probabilities.

Z-Test for the Mean

A z-test is a statistical test used when the population standard deviation is known and the sample size is sufficiently large or the population is normally distributed. It involves comparing a standardized test statistic, calculated from the sample mean, to critical values from the standard normal distribution. This method is common in testing hypotheses about population means, especially in contexts where precision in testing decision criteria is necessary.

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