Transcript Z tests
Practical Statistics Z-Tests There are six statistics that will answer 90% of all questions! 1. 2. 3. 4. 5. 6. Descriptive Chi-square Z-tests Comparison of Means Correlation Regression Z-test are for proportions. This test is so easy…. That it is not even given in some computer programs like SPSS….. Z-test are for proportions. What is the probability that out of 250 customers, 220 would like the service when the usual percent is that 70% (175 out of 250) are satisfied? Z-test are for proportions. What is the probability that out of a random sample of male and female customers, the percent of both men and women who like a new product is the same? Z-test are for proportions. They come in two types: 1. A sample proportion against a hypothesis. Z-test are for proportions. They come in two types: 1. A sample proportion against a hypothesis. 2. Two samples compared to each other. Z-test are for proportions. The standard error for proportions is: SE pq n Where p = freq/total and q = 1 - p Z-test are for proportions. Hence: pt p Z pq n Where p is the hypothesized value, and pt is the proportion found in a sample of size n. Z-test are for proportions. Suppose that XYZ Company believed that 20% of their customers bought 80% of their product (“heavy half”). A sample of 200 customers found that 25% bought 80% of the product. Was the company correct in their estimate? Z-test are for proportions. The test statistic looks like this: Z .25 .20 0.05 177 . (.20 X .80) 0.0283 200 Z-test are for proportions. Z .25 .20 0.05 177 . (.20 X .80) 0.0283 200 Since the test was “two-tailed,” the critical value of Z would be 1.96. Therefore, we would conclude that there is not enough evidence to over-ride the assumption that 20% of the customers bought 80% of the product. Z-test are for proportions. Z .25 .20 0.05 177 . (.20 X .80) 0.0283 200 http://faculty.vassar.edu/lowry/tabs.html#z P = 0.077 Z-test are for proportions. They come in two types: 1. A sample proportion against a hypothesis. 2. Two samples compared to each other. Z-test are for proportions. The test for this case looks like this: ( pt 1 pt 2 ) ( p1 p2 ) Z p1q1 p2 q2 ( ) ( ) n1 n2 Z-test are for proportions. Usually, the test assumes that the two groups Are equal, or: ( pt 1 pt 2 ) 0 Z 1 1 pq ( ) n1 n2 There is a problem here. What is the value of p: ? ( pt 1 pt 2 ) 0 Z 1 1 pq ( ) n1 n2 p is the value of the population proportion, but we usually don’t know that, so p is estimated by the weighted average of the two groups…. p1n1 p2 n2 n1 n2 Suppose that a new product was test marketed in the United States and in Japan. The company hypothesizes that both countries response to the product will be the same. 80% of a sample of 500 said they would buy the product again in the U.S., while 75% of a sample of 200 in Japan said they would buy the product again. Test the hypothesize….. H0 : p1 p2 The test would be: (.80 .75) Z p? But what is p? Since p = 0.80 in the U.S., and 0.75 in Japan, the weighted average is used for p. So: p = ((.8 x 500)+(.75 x 200))/700 = 0.786 The test would be: Z (.80 .75) 1 1 (.785)(.215)( ) 500 200 Z = .05/.0343 = 1.45 The critical value is 1.96; P = 0.147. The null hypothesis cannot be rejected, the U.S. and Japanese customers are assumed to be the same. Questions: 1.Jason works in a building with four floors served by an elevator. He has to use the elevator several times a day to carry heavy material from one floor to the next. He states, only partly in jest, that the elevator hates him because it is never on the floor he is on, and he always has to wait for it to come. To test his hypothesis, he kept track of where the elevator was for one month when he used it. He had 60 recorded elevator uses and 9 times the elevator was at his floor when he needed it. Test Jason’s hypothesis. Questions: 2. Suppose Jason always needs to start at the second floor so, along with the info above, he also kept track of which floor the elevator was actually on when he needed it. He found: first floor: 20 second floor: 9 third floor: 12 and fourth floor: 19 Would the answer to #1 be changed if Jason worked on a different floor? Questions: 3. Women buy 20% more BicTac than men. A new company produces a new version of BicTac supposedly targeted at men. A MR survey of 200 users found that 100 customers of the new product were women. Has the new product made any inroad into the male market? Questions: 4. At your business, all new employees come to work for the same salary. A person (on average) increases their pay at the rate of about 1% a year. The average women has worked at your shop 10 years less than the average male employee. A disgruntled female employee sues you for sexual discrimination because a sample of 100 women showed that the average woman make 13% less than the average man. Does she have statistical evidence for her claim?