Hypothesis Testing Chapter 10 What is the process that determines whether or not your hypothesis regarding data is an accurate one?

Hypothesis and Test procedures 10.1 What are the hypothesis and alternate for an event? Hypothesis—a claim about a population statistic Test of the hypothesis (test procedure)— a method for using sample data to decide between two competing claims the second is the alternate

Null hypothesis—H 0 —the claim initially assumed true Alternate hypothesis—H a —the competing claim —must have same value, but different inequality and be a population statistic --since H o —hypothesized value (given) H a must be > H 0 or < H 0 or ≠ H 0

H o is rejected in favor of H a iff evidence suggests H 0 is false You can only demonstrate strong support for H a by rejecting H 0 NOT rejecting H 0 does NOT mean strong support for H o just a lack of support for H a

Example 1 K mart brand 60 watt lightbulbs state on the package, “Ave. life 1000 hrs.” Let µ denote the true mean life of the Kmart 60 watt bulb. People would be upset if µ were less than 1000 Therefore we test H o : µ = 1000 vs. H a : µ <1000 HW Pg , 2, 4, 6, 8

Errors in Hypothesis testing 10.2 What types of errors exist in hypothesis testing? Type I error—rejecting H 0 when H 0 is true Type II error—failing to rejecting H 0 when H 0 is false – To write the errors think of it as “rejecting H a when H a is true” The only way to be certain an error will not occur is to work with the entire population, which is not always possible.

Level of significance =the probability of a type I error – % of time it is “ok” to reject H 0 when it is true =the probability of a type II error – % of time you fail to reject H 0 when it is false An ideal test has = =0 You can control, you can not control – sometimes if you make too small it makes larger than it needs to be

Typical ‘s.01—used when it is for health reasons or major legal battles.05—the default value.1—when you are just curious

Example 1 A type of lie detector that measures brain waves was developed by a professor of neurobiology at Northwestern University. He said, “it would probably not falsely accuse any innocent people and it would probably pick up 70-90% of guilty people.” Suppose this lie detector test is allowed as evidence and is the sole basis of a decision between the two hypotheses: (not a true test) What are the hypotheses? H 0 = accused is innocent H a = accused is guilty (technically is not innocent) What would happen if a type I error occurred? an innocent person would be found guilty What would happen if a type II error occurred? a guilty person would be found innocent

Example 2 The Associated Press reported that the EPA had warned 819 communities that their tap water contained too much lead. Drinking water is considered unsafe if the mean concentration of lead is 15 parts per billion or greater. The EPA required the cited communities to take corrective action and to monitor lead levels. What are the hypothesis: H 0 : µ ≥ 15 (the lead levels are 15 parts or higher) H a : µ<15 (the lead levels are acceptable) What happens for type I and II errors: Type I it meets the standards when it doesn’t (reject H 0 when its true) Type II it doesn’t meet the standards when it does(failing to reject H 0 when false) What are the consequences of each type of error? Type I—possible health risks Type II—the elimination of a public water source What alpha level would you choose and why?.01—to eliminate health issues associated with lead What if the community only has one water source? Would an alpha of.05 then become acceptable??????? HW PG , 11, 12, 14, 18

Large Sample Hypothesis tests for a population Proportion 10.3 What is a large sample hypothesis test for a population proportion? Test statistic—what you are basing your conclusions on, p,, s Remember: this is also called the observed significance level

Rules for the population Proportion (same as those in Chapters 8 & 9) IF: 1) 2) 3) N must be large (np and n(1-p) ≥ 5 liberal test) 4) the sample must be a simple random sample Then: Therefore: becomes the largest tolerable decimal equivalent of the % that gives the most unusual or most extreme cases generally.01,.05 or.1

Testing h 0 vs. alpha Given H 0 : π = # If H a : π > # upper tail test and use 1- crit z If H a : π < # lower tail test and use crit z If H a : π ≠ # two tail test and use (1-crit z)∙2

Must Haves for the AP TEST and Hypothesis testing Describe the population characteristic associated w/ the hypothesis to be tested ie π, µ, State H 0 State H a Select based on the issue concerned –.01 for health, life threatening issues, some high profile lawsuits –.05 most cases –.1 just curious Write out the z-test with the hypothesized values substituted in Verify that the conditions are acceptable for the test and state them – Simple Random Sample – Sample size is large or the population is normally distributed Compute or state the items needed for the test and calculate the z-score Find the associated critical z and state it State the conclusion in terms of the problem – Reject H 0 if critical z ≤ alpha – Fail to reject H 0 if critical z > alpha – Never state in terms of H a

Example An article on Credit Cards and College students describes the study of payment practices of college students. According to the credit industry, 50% of the college students carry a monthly balance. The Authors randomly polled 310 college students and found 217 carried a balance. Does the sample provide evidence to reject the industries claim to a significance of.05? π= the % of students who carry a balance H 0 : π=.5 H a : π>.5 =.05 n =310 The sample is SRS both np and n(1-p) ≥ 5 310(.7) 310(.3) ≥5 Crit z = 1 It is an upper-tail test therefore Test value = 1 – crit z = 1 -1 = 0 Since 0< Reject H 0 which means the credit card company is not correct and may be extending too much credit to college students

Homework Page 499 to 501 – 20 to 36 multiples of 4

Hypothesis testing for a population Mean 10.4 What is the hypothesis test for a population mean? Similar to the test for a proportion except we use instead of p REMEMBER: if you have a random sample with a large enough n And 1) is known, use the z test Otherwise 2) use the t test and remember to calculate df also remember the t chart tells the opposite of the z chart i.e. the amount above the given t

for the t-test H 0 : µ=# H a : µ># crit t upper tail µ<# 1-crit t lower tail (unless t is negative) µ≠# crit t 2 two tail All other steps are the same

Example 1.Since the sample is large the sample distribution will be approx normal 2.µ=the means weight of the “top 20” starters 3.H 0 : µ = H a : µ < Alpha =.01 (why?????) Since in unknown we will use a t-test Pg 507 Speed, size, and strength are thought to be important factors in football performance. When the top 20 Division I teams were evaluated it was found that the mean weight of starters was 105 kg. A random sample of 33 starting players from Division I schools that were not ranked in the top 20 resulted in a mean of kg with a sample standard deviation of 16.3 kg. Is there sufficient evidence to conclude that the mean weight for non-top 20 starters is less than 105, the known value for the top 20 teams? Conduct the test to a significance level of.01. 6) Since we have a large random sample the test is reasonable 7) n = 33 =103.3 s= 16.3 so 8) This is a lower tail test and since t is negative, we will have the lower tail at df=33-1=32 giving a crit-t of.277 9)Since.277>.01 crit-t > alpha We fail to reject H 0 i.e. there is not sufficient evidence to conclude that the mean weight of the non-top 20 starters is less that that of the top 20

Statistical significance occurs when the value of p or leads to the rejection of H 0 at a particular level of alpha HW pg , 44, 48, 52, 55(show the work)

Power & Probability of type II error 10.5 What is the power and probability of a type II error? The best hypothesis test would: 1)Reduce the likelihood of a type I error and can be accomplished by the choice of alpha 2) Reduce the probability of a type II error and can be accomplished with the power test this is used when we believe there is a statistically different value than the one being presented.

Power of a test— factors: a) the size difference between µ and H 0 the difference the power b) the choice of alpha the alpha the power c) the sample size the n the power

power = prob. of rejecting a false H 0 = not rejecting a false H 0 power = 1- So how do we find the power substitute z in by looking up z=1- and solve for Insert the value we think is true for µ and re- solve for z, =this crit z The power test for a t-test is optional

Example A package delivery service advertises that at least 90% of all packages brought to its office by 9 A.M. for delivery in the same city are delivered by noon that day. Let π denote the proportion of all such packages actually delivered by noon. The hypotheses of interest are π = on time deliveries H 0 : π =.9 H a : π <.9 Where the alternative hypothesis states that the company’s claim is untrue. In random sampling 225 packages delivered, 180 arrived on time. The value of π=.8 represents a substantial departure from the company’s claim. If the hypotheses are tested at level.01 using the sample of n = 225 packages, what is the probability that the departure from H 0 represented by this alternative value will go undetected? Ie what is β

Determining Power 1) π = on time deliveries 2) Ho: π =.9 3) Ha: π <.9 4) 5) We have a random sample and a large # sampled 6) n = 225 for a lower tail test, means p-value of.01 means Z = -2.33, which means p will be 7)Since we found π=.8 8)Crit z or = )Power = =.0233 When π=.8 and alpha =.01 then 2.33% of all samples of 225 will result in a type two error therefore, we do not have a great chance of rejecting a false H 0

Verifying the power test We found So crit z = 0 Since crit z < alpha we reject H 0 Which means our data supports a different H 0 (and we are 97.67% confident that we are correct—last slide) HW pg , 58, 59 show all work

REVIEW Chapter 10 HW Page , 64, 65, 68, 70, 72, 75