9.1 Notes Introduction to Hypothesis Testing

In hypothesis testing there are 2 hypothesis for each problem, the null hypothesis and the alternate hypothesis. Null Hypothesis (H 0 ) – A working hypothesis about the population. *Always uses = i.e. A car dealer claims that the avg. mpg for a certain model is 47. H 0 : μ = 47 mpg Alternate Hypothesis (H 1 /H A ) – Any hypothesis that differs from the null. *Always uses <, >, or ≠. < indicates a left-tailed test, > indicates a right-tailed test ≠ indicates a two-tailed test i.e. We believe the dealer is exaggerating the mpg claim. H 1 : μ < 47 mpg Critical Region Critical Regions Critical Region

Ex. 1 A company manufactures ball bearings for precision machines. The average diameter of a certain type of ball bearing should be 6.0 mm. To check that the average diameter is correct, the company formulates a statistical test. a) What should be the used for H 0 ? H 0 : μ = 6.0 mm b) What should be used for H 1 ? H 1 : μ ≠ 6.0 mm Two-tailed Ex. 2 A package delivery service claims it takes an average of 24 hours to send a package from New York to San Francisco. An independent consumer agency is doing a study to test the truth of this claim. Several complaints have led the agency to suspect that the delivery time is longer than 24 hours. a) What should be the used for H 0 ? H 0 : μ = 24 hours b) What should be used for H 1 ? H 1 : μ > 24 hours right-tailed Hint: an error either way (too large or too small) would be serious

In hypothesis testing there are two possible outcomes, reject the null or fail to reject the null. Reject the Null There is enough evidence in the data to imply that the null is false and the alternate is true. Fail to Reject the Null There is not enough evidence to justify rejecting the null. Neither one of these results are error free. Types of Errors  Type I – We reject the null when in fact the null is true  Type II – We fail to reject the null when in fact the null is false. In order to reduce Type I error, Type II error increases and vice-versa. Relate to court process

Level of Significance (α) – The probability with which we are willing to risk a type I error (reject the null when if fact it is true). Is determined before data is gathered. Used throughout much of the remaining portion of the course. Power of a Test (1 – β) – The probability with which the null is correctly rejected when in fact it is false. Note: β is probability of making a type II error. Hard to calculate and is not related to much in this level of statistics. Some Generalities about α and 1 – β 1.As α increases then 1 – β also increases. 2.Even though an increase in α results in an increase in 1 – β, it also results in a higher probability that we reject the null when in fact it is true. Most people would prefer to accept the null when in fact it is false than to accept the alternate when in fact it is false.

Assignment p. 411 #1-8

Basic Components of a Statistical Test 1. Null Hypothesis H 0, Alternate Hypothesis H 1, and a preset level of significance α If the evidence (sample data) against the H 0 is strong enough, we reject the H 0 and adopt the H 1. The level of significance α is the probability of rejecting H 0 when it is in fact true. 2. Test Statistic and Sampling Distribution (For now we will be focusing mainly on normal and t-student distributions). 3. P-value This is the probability of obtaining a test statistic from the sampling distribution that is as extreme as, or more extreme than the sample test statistic computed from the data under the assumption that H 0 is true. 4. Test Conclusion If P-value α, we do not reject H Interpretation of the test results Give a simple explanation of your conclusions in context of the application.

Ex. 3 The Environmental Protection Agency has been studying Miller Creek regarding ammonia nitrogen concentration. For many years, the concentration has been 2.3 mg/l. However, a new golf course and housing developments are raising concern that the concentration may have changed because of lawn fertilizer. A change either way in ammonia nitrogen concentration can affect plant and animal life in and around the creek. Let x be a random variable representing ammonia nitrogen concentration (in mg/l). Based on recent studies of Miller Creek, we may assume that x has a normal distribution with σ = Recently, a random sample of eight water tests from the creek gave the following x values Test at α = 0.01

a) What is the level of significance? State the null and alternate hypothesis. Will you use a left-tailed, right-tailed, or two-tailed test? b) What sampling distribution will you use? Explain the rationale for your choice of sampling distribution. What is the value of the sample test statistic? c) Find (or estimate) the P-value. Sketch the sampling distribution and show the area corresponding to the P-value. d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data significant at level α? e) State your conclusion in the context of the application.

Assignment P. 413 #9-14