91Ó°ÊÓ

In hypothesis testing, the null hypothesis (\( H_0 \)) is a statement we assume to be true until evidence suggests otherwise. It typically reflects the idea of "no effect" or "no difference." For example, in a quality control test, one might start with a null hypothesis asserting that the mean of a production process is equal to a target value. This sets a benchmark to test against.

Given the examples in your exercise:

• In part (a), \( H_0: \mu = 285 \) means you are assuming the population mean is 285.
• In part (b), \( H_0: \mu = 10.70 \) assumes the population mean is exactly 10.70.
• In part (c), \( H_0: \mu = 147,500 \) implies that the mean income level is hypothesized to be 147,500.

In each case, the goal is to test whether the sample data presents enough evidence to reject this initial assumption.

The test statistic is a crucial part of hypothesis testing. It helps to determine how far the sample statistic (like the sample mean) is from the parameter specified in the null hypothesis. The test statistic for a mean in a t-test is calculated by:\[ t = \frac{\bar{x} - \mu}{\frac{s}{\sqrt{n}}}\]Here's what each symbol represents:

• \( \bar{x} \) is the sample mean.
• \( \mu \) is the population mean under the null hypothesis.
• \( s \) is the sample's standard deviation.
• \( n \) is the sample size.

This formula essentially measures the difference between the sample mean and the population mean, standardized by the standard error. By plugging in numbers from actual data, you can calculate the exact test statistic value to further evaluate the hypothesis.

The t-distribution is a probability distribution that is used in hypothesis testing, particularly when dealing with small sample sizes or when the population standard deviation is unknown. Unlike the normal distribution, the t-distribution is broader and has heavier tails. As the sample size increases, it becomes similar to a normal distribution.

In hypothesis testing, if the test statistic falls into the critical region of the t-distribution (determined by the chosen significance level \( \alpha \)), we then have enough evidence to reject the null hypothesis. The critical values from the t-distribution depend on:

• The degrees of freedom, which is often the sample size minus one (\( n - 1 \)).
• The significance level (\( \alpha \)), which dictates how much risk of error you're willing to accept.

For example, in part (b) with a two-tailed test of \( \alpha = 0.01 \), you'll find two critical values corresponding to both tails of the distribution. The t-distribution is key because it adjusts for the sample's size and variability when no exact population parameters are known.

The sample mean (\( \bar{x} \)) is a fundamental concept in statistics as it provides an estimate of the population mean. It is calculated as the average of all observations in a sample. In hypothesis testing, the sample mean helps us understand how close or far the sample observations are on average from the hypothesized population mean.

For example, in part (a), your sample mean is 267.80, a value used to test against the null hypothesis of \( \mu = 285 \). Similarly, part (b) involves a sample mean of 12.025 which is tested against \( \mu = 10.70 \). The exercise uses the sample mean to calculate the test statistic which evaluates the plausibility of the null hypothesis.

Understanding the sample mean provides insight into the general trend of your data and it forms the base for further statistical analysis.