Hypothesis Testing. Lecture 10

Transcription

1 Lecture 10 Hypothesis Testing A hypothesis is a conjecture about the distribution of some random variables. For example, a claim about the value of a parameter of the statistical model. There are two types of hypotheses: The null hypothesis, The alternative hypothesis, want to show., is the current belief., is your belief; it is what you Examples: Each of the following situations requires a significance test about a population mean. State the appropriate null hypothesis and alternative hypothesis in each case. (a) The mean area of the several thousand apartments in a new development is advertised to be 1250 square feet. A tenant group thinks that the apartments are smaller than advertised. They hire an engineer to measure a sample of apartments to test their suspicion.

2 (b) Larry's car consume on average 32 miles per gallon on the highway. He now switches to a new motor oil that is advertised as increasing gas mileage. After driving 3000 highway miles with the new oil, he wants to determine if his gas mileage actually has increased. (c) The diameter of a spindle in a small motor is supposed to be 5 millimeters. If the spindle is either too small or too large, the motor will not perform properly. The manufacturer measures the diameter in a sample of motors to determine whether the mean diameter has moved away from the target.

3 Guidelines for Hypothesis testing Hypothesis testing is a proof by contradiction. The testing process has four steps: Step 1: Assume is true. Step 2: Use statistical theory to make a statistic (function of the data) that includes. This statistic is called the test statistic. Step 3: Find the probability that the test statistic would take a value as extreme or more extreme than that actually observed. Think of this as: probability of getting our sample assuming is true. Step 4: If the probability we calculated in step 3 is high it means that the sample is likely under and so we have no evidence against. If the probability is low, there are two possibilities: - we observed a very unusual event, or - our assumption is wrong

4 Test Statistic The test is based on a statistic that estimates the parameter that appears in the hypotheses. Usually this is the same estimate we would use in a confidence interval for the parameter. When is true, we expect the estimate to take a value near the parameter value specified in. Values of the estimate far from the parameter value specified by give evidence against. The alternative hypothesis determines which directions count against. A test statistic measures compatibility between the null hypothesis and the data. To assess how far the estimate is from the parameter, standardize the estimate. In many common situations the test statistics has the form

5 Example: An air freight company wishes to test whether or not the mean weight of parcels shipped on a particular root exceeds 10 pounds. A random sample of 49 shipping orders was examined and found to have average weight of 11 pounds. Assume that the standard deviation of the weights is 2.8 pounds. Solution:

6 Graphical Representation Suppose we want to test a set of hypotheses concerning a parameter based on a random sample. vs is the estimate of our parameter. Rejection Region (RR) is the specified values of the test statistics for which we reject. The probability that defines the critical region is called the size of the test or level of the significance of the test and is denoted by α.

7 Example: The hourly wages in a particular industry are normally distributed with mean $13.20 and standard deviation $2.50. A company employs 40 workers paying them an average of $12.20 per hour. Can this company be accused of paying substandard wages? Use. Solution:

8 Decision Errors When we perform a statistical test we hope that our decision will be correct, but sometimes it will be wrong. There are two possible errors that can be made in hypothesis test. Definition: The error made by rejecting the null hypothesis when in fact is true is called a type I error. The error made by failing to reject the null hypothesis when in fact is false is called a type II error. Note: The level of significance of the test is also the probability of type I error, denoted by, i.e. The probability of a type II error is denoted by. Example: An experimenter has prepared a drug dosage level that she claims will induce sleep for 80% of people suffering from insomnia. In an attempt to disprove her claim, we administer her prescribed dosage to 20 insomniacs and observe X, the number of people for whom the drug dose induces sleep. We wish to test vs. Assume.

9 P-value Definition: The probability, assuming is true, that the test statistic would take a value as extreme or more extreme than that actually observed is called the P-value of the test. The smaller the P-value, the stronger the evidence against provided by the data. Guideline for how small is small : P-value > 0.1 provides no evidence against < P-value < 0.1 provides weak evidence against < P-value < 0.05 provides moderated evidence against. P-value < 0.01 provides strong evidence against. We can compare the P-value we calculate with a fixed value that we regard as decisive. The decisive value of P is called the significance level (this is our ). Most common values for are 0.1, 0.05, If the P-value is as small or smaller than, we say that the data are statistically significant at level. In other words, the P-value is the smallest level of significance for which the null hypothesis should be rejected.

10 Example: 85% of the general public is right-handed. A survey of 300 chief executive officers of large corporations found that 95% were right-handed. Is this difference in percentages statistically significant? Use. Find the P-value for the test. Solution:

11 Tests for a Population Mean ( is known) where is the specified value of.

12

13 Example: In 1999, it was reported that the mean serum cholesterol level for female undergraduates was 168 mg/dl with a standard deviation of 27 mg/dl. A recent study at Baylor University investigated the lipid levels in a cohort of sedentary university students. The mean total cholesterol level among n = 71 females was. Is this evidence that cholesterol levels of sedentary students differ from the previously reported average? Solution:

14 Tests for a Population Mean ( is unknown)

15 Recall: (one-sample t CI) Example: Founded in 1998, Telephia provides a wide variety of information on cellular phone use. In 2006, Telaphia reported that, on average, United Kingdom (U.K.) subscribers with thirdgeneration technology (3G) phones spent an average of 8.3 hours per month listening to full-track music on their cell phones. Suppose we want to determine a 95% CI for the U.S. average and draw the following random sample of size 8 from the U.S. population of 3G subscribers: The sample mean is and the standard deviation s = 3.63 with degrees of freedom n - 1 = 7.

16

17 Example: Suppose that, for the U.S. data in example before we want to test whether the U.S. average is different from the reported U.K. average.

18 Power The ability of a test to detect that is false is measured by the probability that the test will reject when an alternative is true. The higher this probability is, the more sensitive the test is. Definition: The probability that a fixed size when is false is called the power of the test. test will reject A powerful test has a large probability of rejecting false. when it is Example: Can a 6-month exercise program increase the total body bone mineral content (TBBMC) of young women? A team of researchers is planning a study to examine this question. Based on the results of a previous study, they are willing to assume that for the percent change in TBBMC over the 6-month period. A change in TBBMC of 1% would be considered important, and the researchers would like to have a reasonable chance of detecting a change this large or larger. Is 25 subjects a large enough sample for this project? Three steps to find the power of the test: 1. State,, the particular alternative we want to detect, and the significance level. 2. Find the values of (or other estimates) that will lead to reject. 3. Calculate the probability of observing these values of when the alternative is true.

19

20 How to increase the power?

21 Back to Error Probabilities Example: The mean outer diameter of a skateboard bearing is supposed to be millimeters (mm). The outer diameters vary Normally with standard deviation mm. When a lot of bearings arrives, the skateboard manufacturer takes an SRS of 5 bearings from the lot and measures their outer diameters. The manufacturer rejects the bearings if the sample mean diameter is significantly different from 22 at the 5% significance level. Suppose the producer and the manufacturer agree that a lot of bearings with mean mm away from 22 should be rejected.

22 Significance and Type I error: The significance level of any fixed level test is the probability of a Type I error. That is, is the probability that the test will reject when is in fact true. Power and Type II error: The power of a fixed level test to detect a particular alternative is 1 minus the probability of a Type II error for that alternative.

23 Comparing Two Means Assume we have two populations of interest, each with unknown mean. Choose an SRS of size from one normal population having mean and standard deviation and an independent SRS of size from another normal population having mean and standard deviation. The estimate of the difference in the population means is where and are sample means. Distribution of :

24 Example: A fourth-grade class has 12 girls and 8 boys. The children s heights are recorded on their 10 th birthdays. Based on information from the National Health and Nutrition Examination Survey, the heights (in inches) of 10-year-old girls are distributed Normally with mean 56.8 and standard deviation 2.7 and the heights (in inches) of 10-year-old boys are distributed Normally with mean 55.7 and standard deviation 3.8. Assume that the heights of the students in the class are random samples from the populations. What is the probability that the girls average height is greater than the boys average height? Solution:

25 Here we know and, which is quite rare. So in general, there are two ways to compare the means of two normal populations. This is due to the fact that there are two distinct possibilities: 1. and are unknown and equal. 2. and are unknown and unequal.

26 Comparing Two Mean: Variances Unequal Assume and are unknown. We estimate them by and.

27 Example: An educator believes that new directed reading activities in the classroom will help elementary school pupils improve some aspects of their reading ability. She arranges for a third-grade class of 21 students to take part in these activities for an eight-week period. A control classroom of 23 third-graders follows the same curriculum without the activities. At the end of the eight weeks, all students are given a Degree of Reading Power (DRP) test, which measures the aspects of reading ability that the treatment is designed to improve. The data appear in the table below:

28

29 Two-Sample t CI: Choose an SRS of size from a Normal population with unknown mean and an independent SRS of size from another Normal population with unknown mean. A ( ) CI for is given by ( ) where is the value for density curve with area between and. The value of the degrees of freedom k is approximated by software or we use the smaller of and. Example: How much improvement?

30 Comparing Two Means: Variances Equal (Pooled Test) Suppose we have two Normal populations with the same variances:, is unknown.

31 The pooled two-sample t procedures: Choose an SRS of size from a Normal population with unknown mean and an independent SRS of size from another Normal population with unknown mean. A ( ) CI for is given by ( ) where is the value for density curve with area between and., compute the pooled two- To test the hypothesis sample t statistic In terms of a random variable T having the the P-value for a test of against distribution,

32 Example: Does increasing the amount of calcium in our diet reduce blood pressure? Examination of a large sample of people revealed a relationship between calcium intake and blood pressure, but such observational studies do not establish causation. A randomized comparative experiment gave one group of 10 people a calcium supplement for 12 weeks. The control group of 11 people received a placebo that appeared identical. Table below gives the seated systolic blood pressure for all subjects at the beginning and end of 12-week period, in millimeters of mercury. The table also shows the decrease of each subject. An increase appears as a negative entry.

33 Does increase calcium reduce blood pressure?

34 How different are the calcium and placebo groups?