Introduction to Inference Tests of Significance

Proof

Proof

Definitions A test of significance is a method for using sample data to decide between two competing claims about a population characteristic. The null hypothesis, denoted by H 0, says that there is no effect or no change to a claim assumed to be true (i.e. H 0 :  1000). The alternative hypothesis, denoted by H a, is the competing claim (i.e. H a :  1000).

Chrysler Concord H 0 :  H a : 

Phrasing our decision In justice system, what is our null and alternative hypothesis? H 0 : defendant is innocent H a : defendant is guilty What does the jury state if the defendant wins? Not guilty Why?

Phrasing our decision H 0 : defendant is innocent H a : defendant is guilty If we have the evidence: –We reject the belief the defendant is innocent because we have the evidence to believe the defendant is guilty. If we don’t have the evidence: –We fail to reject the belief the defendant is innocent because we do not have the evidence to believe the defendant is guilty.

Chrysler Concord H 0 :  H a :  p-value =.0134 We reject H 0 since the probability is so small there is enough evidence to believe the mean Concord time is greater than 8 seconds.

K-mart light bulb H 0 :  H a :  p-value =.1078 We fail to reject H 0 since the probability is not very small there is not enough evidence to believe the mean lifetime is less than 1000 hours.

Remember: Inference procedure overview State the procedure Define any variables Establish the conditions (assumptions) Use the appropriate formula Draw conclusions

Test of Significance Example 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 the claim. Several complaints have led the agency to suspect that the delivery time is longer than 24 hours. Assume that the delivery times are normally distributed with standard deviation (assume  for now) of 2 hours. A random sample of 25 packages has been taken.

Example 1 test of significance  = true mean delivery time H o :  = 24 H a :  > 24 Given a random sample Given a normal distribution Safe to infer a population of at least 250 packages

Example 1 (look, don’t copy)

Example 1 let  =.05 test of significance  = true mean delivery time H o :  = 24 H a :  > 24 Given a random sample Given a normal distribution Assume a population of at least 250 packages

Example 1 test of significance  = true mean delivery time H o :  = 24 H a :  > 24 Given a random sample Given a normal distribution Assume a population of at least 250 packages We reject H o. Since p-value<  there is enough evidence to believe the delivery time is longer than 24 hours.

Wording of conclusion revisit If I believe the statistic is just too extreme and unusual (P-value <  ), I will reject the null hypothesis. If I believe the statistic is just normal chance variation (P-value >  ), I will fail to reject the null hypothesis. We reject fail to reject H o, since the p-value< , there is p-value> , there is not enough evidence to believe…(H a in context…)

Example 4 test of significance  = true mean distance H o :  = 340 H a :  > 340 Assume a random sample Assume normal distribution Safe to infer a population of at least 100 missiles. We fail to reject H o. Since p-value>  there is not enough evidence to believe the mean distance traveled is more than 340 miles.

Familiar transition What happened on day 2 of confidence intervals involving mean and standard deviation? Switch from using z-scores to using the t- distribution. What changes occur in the write up?

Example 4 test of significance  = true mean distance H o :  = 340 H a :  > 340 Assume a random sample Assume normal distribution Safe to infer a population of at least 100 missiles. We fail to reject H o. Since p-value>  there is not enough evidence to believe the mean distance traveled is more than 340 miles, but…

Example 4 t-test  = true mean distance H o :  = 340 H a :  > 340 Assume a random sample Assume normal distribution Safe to infer a population of at least 100 missiles. We fail to reject H o. Since p-value>  there is not enough evidence to believe the mean distance traveled is more than 340 miles, but…

Example 4 t-test  = true mean distance H o :  = 340 H a :  > 340 Assume a random sample Assume normal distribution Safe to infer a population of at least 100 missiles. We fail to reject H o. Since p-value>  there is not enough evidence to believe the mean distance traveled is more than 340 miles, but

Example 4 t-test  = true mean distance H o :  = 340 H a :  > 340 Assume a random sample Assume normal distribution Safe to infer a population of at least 100 missiles. We fail to reject H o. Since p-value>  there is not enough evidence to believe the mean distance traveled is more than 340 miles, but

t-chart

Example 4 t-test  = true mean distance H o :  = 340 H a :  > 340 Assume a random sample Assume normal distribution Safe to infer a population of at least 100 missiles. We fail to reject H o. Since p-value>  there is not enough evidence to believe the mean distance traveled is more than 340 miles, but

Example 4 t-test  = true mean distance H o :  = 340 H a :  > 340 Assume a random sample Assume normal distribution Safe to infer a population of at least 100 missiles. We fail to reject H o. Since p-value>  there is not enough evidence to believe the mean distance traveled is more than 340 miles, but

1 proportion z-test p = true proportion pure short H o : p =.25 H a : p =.25 Given a random sample. np = 1064(.25) > 10 n(1–p) = 1064(1–.25) > 10 Sample size is large enough to use normality Safe to infer a population of at least 10,640 plants. We fail to reject H o. Since p-value>  there is not enough evidence to believe the proportion of pure short is different than 25%.

Choosing a level of significance How plausible is H 0 ? If H 0 represents a long held belief, strong evidence (small  ) might be needed to dissolve the belief. What are the consequences of rejecting H 0 ? The choice of  will be heavily influenced by the consequences of rejecting or failing to reject.

Errors in the justice system Actual truth Jury decision GuiltyNot guilty Guilty Not guilty Correct decision Type I error Type II error

“No innocent man is jailed” justice system Actual truth Jury decision GuiltyNot guilty Guilty Not guilty Type I error Type II error smaller larger

“No guilty man goes free” justice system Actual truth Jury decision GuiltyNot guilty Guilty Not guilty Type I error Type II error smaller larger

Errors in the justice system Actual truth Jury decision GuiltyNot guilty Guilty Not guilty Correct decision Type I error Type II error (H a true)(H 0 true) (reject H 0 ) (fail to reject H 0 )

Type I and Type II errors If we believe H a when in fact H 0 is true, this is a type I error. If we believe H 0 when in fact H a is true, this is a type II error. Type I error: if we reject H 0 and it’s a mistake. Type II error: if we fail to reject H 0 and it’s a mistake. APPLET

Type I and Type II example A distributor of handheld calculators receives very large shipments of calculators from a manufacturer. It is too costly and time consuming to inspect all incoming calculators, so when each shipment arrives, a sample is selected for inspection. Information from the sample is then used to test Ho: p =.02 versus Ha: p <.02, where p is the true proportion of defective calculators in the shipment. If the null hypothesis is rejected, the distributor accepts the shipment of calculators. If the null hypothesis cannot be rejected, the entire shipment of calculators is returned to the manufacturer due to inferior quality. (A shipment is defined to be of inferior quality if it contains 2% or more defectives.)

Type I and Type II example Type I error: We think the proportion of defective calculators is less than 2%, but it’s actually 2% (or more). Consequence: Accept shipment that has too many defective calculators so potential loss in revenue.

Type I and Type II example Type II error: We think the proportion of defective calculators is 2%, but it’s actually less than 2%. Consequence: Return shipment thinking there are too many defective calculators, but the shipment is ok.

Type I and Type II example Distributor wants to avoid Type I error. Choose  =.01 Calculator manufacturer wants to avoid Type II error. Choose  =.10

Concept of Power Definition? Power is the capability of accomplishing something… The power of a test of significance is…

Power Example In a power generating plant, pressure in a certain line is supposed to maintain an average of 100 psi over any 4 - hour period. If the average pressure exceeds 103 psi for a 4 - hour period, serious complications can evolve. During a given 4 - hour period, thirty random measurements are to be taken. The standard deviation for these measurements is 4 psi (graph of data is reasonably normal), test H o :  = 100 psi versus the alternative “new” hypothesis  = 103 psi. Test at the alpha level of.01. Calculate a type II error and the power of this test. In context of the problem, explain what the power means.

Type I error and   is the probability that we think the mean pressure is above 100 psi, but actually the mean pressure is 100 psi (or less)

Type I error and 

Type II error and 

 is the probability that we think the mean pressure is 100 psi, but actually the pressure is greater than 100 psi.

Power?

For a sample size of 30, there is a.9495 probability that this test of significance will correctly detect if the pressure is above 100 psi.

Concept of Power The power of a test of significance is the probability that the null hypothesis will be correctly rejected. Because the true value of  is unknown, we cannot know what the power is for , but we are able to examine “what if” scenarios to provide important information. Power = 1 – 

Effects on the Power of a Test The larger the difference between the hypothesized value and the true value of the population characteristic, the higher the power. The larger the significance level, , the higher the power of the test. The larger the sample size, the higher the power of the test. APPLET