# 9: Hypothesis Testing with One Sample

- Page ID
- 20708

One job of a statistician is to make statistical inferences about populations based on samples taken from the population. Confidence intervals are one way to estimate a population parameter. Another way to make a statistical inference is to make a decision about a parameter. For instance, a car dealer advertises that its new small truck gets 35 miles per gallon, on average. A tutoring service claims that its method of tutoring helps 90% of its students get an A or a B. A company says that women managers in their company earn an average of $60,000 per year.

- 9.1: Prelude to Hypothesis Testing
- A statistician will make a decision about claims via a process called "hypothesis testing." A hypothesis test involves collecting data from a sample and evaluating the data. Then, the statistician makes a decision as to whether or not there is sufficient evidence, based upon analysis of the data, to reject the null hypothesis.

- 9.2: Null and Alternative Hypotheses
- The actual test begins by considering two hypotheses. They are called the null hypothesis and the alternative hypothesis. These hypotheses contain opposing viewpoints. Since the null and alternative hypotheses are contradictory, you must examine evidence to decide if you have enough evidence to reject the null hypothesis or not.

- 9.3: Outcomes and the Type I and Type II Errors
- 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.

- 9.4: Distribution Needed for Hypothesis Testing
- 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.

- 9.5: Rare Events, the Sample, Decision and Conclusion
- 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.

- 9.6: Additional Information and Full Hypothesis Test Examples
- The hypothesis test itself has an established process. This can be summarized as follows: Determine H0 and Ha. Remember, they are contradictory. Determine the random variable. Determine the distribution for the test. Draw a graph, calculate the test statistic, and use the test statistic to calculate the p-value. (A z-score and a t-score are examples of test statistics.) Compare the preconceived α with the p-value, make a decision (reject or do not reject H0), and write a clear conclusion.

- 9.7: Large Sample Tests for a Population Mean
- In this section we describe and demonstrate the procedure for conducting a test of hypotheses about the mean of a population in the case that the sample size n is at least 30

- 9.8: The Observed Significance of a Test
- The conceptual basis of our testing procedure is that we reject the null hypothesis only if the data that we obtained would constitute a rare event if the null hypothesis were actually true. The level of significance α specifies what is meant by “rare.” The observed significance of the test is a measure of how rare the value of the test statistic that we have just observed would be if the null hypothesis were true.

- 9.9: Small Sample Tests for a Population Mean
- Previous hypotheses testing for population means was described in the case of large samples. The statistical validity of the tests was insured by the Central Limit Theorem, with essentially no assumptions on the distribution of the population. When sample sizes are small, as is often the case in practice, the Central Limit Theorem does not apply. One must then impose stricter assumptions on the population to give statistical validity to the test procedure.

- 9.10: Large Sample Tests for a Population Proportion
- Both the critical value approach and the p-value approach can be applied to test hypotheses about a population proportion.

- 9.E: Hypothesis Testing with One Sample (Exercises)
- These are homework exercises to accompany the Textmap created for "Introductory Statistics" by OpenStax.

- 9.E: Testing Hypotheses (Exercises)
- These are homework exercises to accompany the Textmap created for "Introductory Statistics" by Shafer and Zhang.

## Contributors and Attributions

Barbara Illowsky and Susan Dean (De Anza College) with many other contributing authors. Content produced by OpenStax College is licensed under a Creative Commons Attribution License 4.0 license. Download for free at http://cnx.org/contents/30189442-699...b91b9de@18.114.