Introduction to Inference Tests of Significance

Proof

Proof

Definitions A test of significance is a method for using sample data to make a decision about a population characteristic. The null hypothesis, written H 0, is the starting value for the decision (i.e. H 0 :  1000). The alternative hypothesis, written H a, states what belief/claim we are trying to determine if statistically significant (H a :  1000).

Examples Chrysler Concord –H 0 :  –H a :  K-mart –H 0 :  –H a : 

Chrysler 8

K-mart 1000

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 Safe to infer 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 Safe to infer 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 3 test of significance  = true mean distance H o :  = 340 H a :  > 340 Given random sample Given normally distributed. 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 3 test of significance  = true mean distance H o :  = 340 H a :  > 340 Given random sample. Given normally distributed. 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.

Example 3 t-test  = true mean distance H o :  = 340 H a :  > 340 Given random sample. Given normally distributed. 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.

Example 3 t-test  = true mean distance H o :  = 340 H a :  > 340 Given random sample Given normally distributed. 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.

Example 3 t-test  = true mean distance H o :  = 340 H a :  > 340 Given random sample. Given normally distributed. 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.

t-chart

Example 3 t-test  = true mean distance H o :  = 340 H a :  > 340 Given random sample. Given normally distributed. 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.

Example 3 t-test  = true mean distance H o :  = 340 H a :  > 340 Given random sample. Given normally distributed. 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.