Chapter 8: Hypothesis Testing with One Sample
Chapter 8 Review
8.1 Chapter Review
In a hypothesis test, sample data is evaluated in order to arrive at a decision about some type of claim. If certain conditions about the sample are satisfied, then the claim can be evaluated for a population. In a hypothesis test, we:
Formula Review
[latex]H_{0}[/latex] and [latex]H_{a}[/latex] are contradictory.
| If [latex]H_{0}[/latex] has: | equal (=) | greater than or equal to (≥) | less than or equal to (≤) |
| then [latex]H_{a}[/latex] has: | not equal (≠) or greater than (>) or less than (<) | less than (<) | greater than (>) |
If [latex]\alpha \le \text{p-value}[/latex], then do not reject [latex]H_{0}[/latex].
If [latex]\alpha > \text{p-value}[/latex], then reject [latex]H_{0}[/latex].
[latex]\alpha[/latex] is preconceived. Its value is set before the hypothesis test starts. The p-value is calculated from the data.
8.2 Chapter Review
In every hypothesis test, the outcomes are dependent on a correct interpretation of the data. Incorrect calculations or misunderstood summary statistics can yield errors that affect the results. A Type I error occurs when a true null hypothesis is rejected. A Type II error occurs when a false null hypothesis is not rejected.
The probabilities of these errors are denoted by the Greek letters [latex]\alpha[/latex] and [latex]\beta[/latex], for a Type I and a Type II error respectively. The power of the test, [latex]1 - \beta[/latex], quantifies the likelihood that a test will yield the correct result of a true alternative hypothesis being accepted. A high power is desirable.
Formula Review
[latex]\alpha=[/latex] probability of a Type I error = P(Type I error) = probability of rejecting the null hypothesis when the null hypothesis is true.
[latex]\beta=[/latex] probability of a Type II error = P(Type II error) = probability of not rejecting the null hypothesis when the null hypothesis is false.
8.3 Chapter Review
In order for a hypothesis test’s results to be generalized to a population, certain requirements must be satisfied.
When testing for a single population mean:
- A Student’s t-test should be used if the data come from a simple, random sample and the population is approximately normally distributed, or the sample size is large, with an unknown standard deviation.
- The normal test will work if the data come from a simple, random sample and the population
is approximately normally distributed, or the sample size is large, with a known standard
deviation.
When testing a single population proportion use a normal test for a single population proportion if the data comes from a simple, random sample, fill the requirements for a binomial distribution, and the mean number of success and the mean number of failures satisfy the conditions: [latex]np > 5[/latex] and [latex]nq > n[/latex] where [latex]n[/latex] is the sample size, [latex]p[/latex] is the probability of a success, and [latex]q[/latex] is the probability of a failure.
Formula Review
If there is no given preconceived [latex]\alpha[/latex], then use [latex]\alpha = 0.05[/latex].
- Single population mean, known population variance (or standard deviation): Normal
test. - Single population mean, unknown population variance (or standard deviation): Student’s t-test.
- Single population proportion: Normal test.
- For a single population mean, we may use a normal distribution with the following mean and standard deviation. Means: [latex]\mu ={\mu }_{\overline{x}}[/latex] and [latex]{\sigma }_{\overline{x}}=\frac{{\sigma }_{x}}{\sqrt{n}}[/latex]
- A single population proportion, we may use a normal distribution with the following mean and standard deviation. Proportions: [latex]\mu = p[/latex] and [latex]\sigma =\sqrt{\frac{pq}{n}}[/latex].
8.4 Chapter Review
When the probability of an event occurring is low, and it happens, it is called a rare event. Rare events are important to consider in hypothesis testing because they can inform your willingness not to reject or to reject a null hypothesis. To test a null hypothesis, find the p-value for the sample data and graph the results. When deciding whether or not to reject the null the hypothesis, keep these two parameters in mind:
8.5 Chapter Review
The hypothesis test itself has an established process. This can be summarized as follows:
Notice that in performing the hypothesis test, you use [latex]\alpha[/latex] and not [latex]\beta[/latex]. [latex]\beta[/latex] is needed to help determine the sample size of the data that is used in calculating the p-value. Remember that the quantity [latex]1 - \beta[/latex] is called the Power of the Test. A high power is desirable. If the power is too low, statisticians typically increase the sample size while keeping [latex]\alpha[/latex] the same. If the power is low, the null hypothesis might not be rejected when it should be.
