The mean income per person in the United States is 50,000,...

The mean income per person in the United States is $\$ 50,000,$ and the distribution of incomes follows a normal distribution. A random sample of 10 residents of Wilmington, Delaware, had a mean of $\$ 60,000$ with a standard deviation of $\$ 10,000 .$ At the .05 level of significance, is that enough evidence to conclude that residents of Wilmington, Delaware, have more income than the national average?

Key Concepts

Hypothesis Testing

Hypothesis testing is a statistical method used to make decisions or inferences about a population parameter based on sample data. It involves formulating a null hypothesis, which represents a status quo or a claim to be tested, and an alternative hypothesis, which represents the assertion that the sample data supports. The process then involves determining whether the observed data falls within a range that would lead to the rejection of the null hypothesis with a known probability of making a Type I error.

Null and Alternative Hypotheses

The formulation of null and alternative hypotheses is a key concept in hypothesis testing. The null hypothesis typically states that there is no effect or difference, or that the parameter is equal to a specific value, while the alternative hypothesis presents a claim that contradicts the null, such as the parameter being greater than or less than that specific value. In testing claims about a population mean, the structure of these hypotheses guides the direction and type of statistical test to be used.

Significance Level

The significance level, often denoted by alpha (?), is the threshold probability for making a Type I error—incorrectly rejecting a true null hypothesis. It determines the critical region or the rejection region for the test statistic. A significance level of 0.05 means that there is a 5% risk of concluding that a difference or effect exists when in fact there is none.

One-Sample t-Test

The one-sample t-test is used when testing hypotheses about the mean of a normally distributed population when the population standard deviation is unknown and the sample size is relatively small. This test employs the t-distribution to estimate the probability of observing a sample mean at least as extreme as the one obtained, given that the null hypothesis is true. It accounts for additional uncertainty inherent in estimating the population standard deviation from the sample.

Test Statistic

The test statistic is a standardized value calculated from sample data during a hypothesis test. It measures how many standard errors the sample mean deviates from the hypothesized population mean. For a one-sample t-test, the test statistic follows a t-distribution with degrees of freedom equal to the sample size minus one, which is then used to determine the p-value and the decision regarding the null hypothesis.

Transcript

Please give Ace some feedback