Introduction to Hypothesis Testing 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 (H0) – i.e. A car dealer claims that the avg. mpg for a certain model is 47. Alternate Hypothesis (H1/HA) – < indicates > indicates ≠ indicates i.e. We believe the dealer is exaggerating the mpg claim.

Ex. 1 A company manufactures ball bearings for precision machines 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 H0? b) What should be used for H1? 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 H0?

Neither one of these results are error free. Types of Errors Type I – In hypothesis testing there are two possible outcomes, reject the null or fail to reject the null. Reject the Null Fail to Reject the Null Neither one of these results are error free. Types of Errors Type I – Type II – 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 As α increases then 1 – β also increases. 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. 412 #1-8

Basic Components of a Statistical Test 1. _____ Hypothesis H0 , ____________ Hypothesis H1 , and a preset _____ ____________________ α If the evidence (sample data) against the H0 is strong enough, we ________________________. The level of significance α is the probability of ______________________________________. 2. Test Statistic and Sampling Distribution (For now we will be focusing mainly on _________ and ___________ distributions). 3. P-value This is the probability of obtaining a test statistic from the sampling distribution that is _____________, or _________________than the sample test statistic computed from the data under the assumption that H0 is true. 4. Test Conclusion If P-value ________, we reject H0 and say that the data are significant at level α. If P-value ________, we do not reject H0. 5. Interpretation of the test results Give a simple explanation of your conclusions in context of the application.

Ex. 3 Rosie is an aging sheep dog in Montana who gets regular check-ups from her owner, the local veterinarian. Let x be a random variable that represents Rosie’s resting heart rate (in beats per minute). From past experience, the vet knows that x has a normal distribution with σ = 12. The vet checked the Merck Veterinary Manual and found that for dogs of this breed, μ = 115 beats per minute. Over the past six weeks, Rosie’s heart rate (beats/minute) measured 93 109 110 89 112 117 The vet is concerned that Rosie’s heart rate is falling below normal. Do the data indicate that this is the case? Test at α = 0.05

a) What is the level of significance 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.

Ex. 4 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 = 0.30. Recently, a random sample of eight water tests from the creek gave the following x values. 2.1 2.5 2.2 2.8 3.0 2.2 2.4 2.9 Test at α = 0.01

a) What is the level of significance 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