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HYPOTHESIS TESTSABOUT THE MEAN ANDPROPORTIONPrem Mann Introductory Statistics 9 ECopyright 2015 John Wiley Sons All rights reserved .

Opening ExampleWill you graduate from college with debt If yes how much money do you thinkyou will owe Do you know that students who graduated from public andnonprofit colleges in 2013 with loans had an average debt of 28 400 Theaverage debt of the class of 2013 varied a great deal from state to state with.

the highest average debt for students who graduated from colleges in NewHampshire at 32 795 and the lowest average for students who graduated fromcolleges in New Mexico at 18 656 See Case Study 9 1 Prem Mann Introductory Statistics 9 ECopyright 2015 John Wiley Sons All rights reserved .

9 1 Hypothesis Tests An Introduction Two Hypotheses Rejection and Nonrejection Regions Two Types of Errors Tails of a Test.

Prem Mann Introductory Statistics 9 ECopyright 2015 John Wiley Sons All rights reserved Two HypothesesDefinitionA null hypothesis is a claim or statement about a.

population parameter that is assumed to be true until it isdeclared false Prem Mann Introductory Statistics 9 ECopyright 2015 John Wiley Sons All rights reserved Two Hypotheses.

DefinitionAn alternative hypothesis is a claim about a populationparameter that will be declared true if the null hypothesis isdeclared to be false Prem Mann Introductory Statistics 9 E.

Copyright 2015 John Wiley Sons All rights reserved Rejection and Nonrejection RegionsFigure 9 1 Nonrejection and Rejection Regions forthe Court CasePrem Mann Introductory Statistics 9 E.

Copyright 2015 John Wiley Sons All rights reserved Two Types of ErrorsTable 9 1 Four Possible Outcomes for a Court CasePrem Mann Introductory Statistics 9 ECopyright 2015 John Wiley Sons All rights reserved .

Two Types of ErrorsDefinitionA Type I error occurs when a true null hypothesis is rejected The value of represents the probability of committing thistype of error that is .

P H0 is rejected H0 is true The value of represents the significance level of the test Prem Mann Introductory Statistics 9 ECopyright 2015 John Wiley Sons All rights reserved Two Types of Errors.

DefinitionA Type II error occurs when a false null hypotheses is notrejected The value of represents the probability ofcommitting a Type II error that is P H0 is not rejected H0 is false .

The value of 1 is called the power of the test Itrepresents the probability of not making a Type II error Prem Mann Introductory Statistics 9 ECopyright 2015 John Wiley Sons All rights reserved Table 9 2 Four Possible Outcomes for a Test of.

HypothesisPrem Mann Introductory Statistics 9 ECopyright 2015 John Wiley Sons All rights reserved Tails of a TestDefinition.

A two tailed test has rejection regions in both tails a left tailed test has the rejection region in the left tail and aright tailed test has the rejection region in the right tail ofthe distribution curve Prem Mann Introductory Statistics 9 E.

Copyright 2015 John Wiley Sons All rights reserved A Two Tailed Test According to the U S Bureau of Labor Statistics workers inthe United States who were employed earned an average of 47 230 a year in 2014 Suppose an economist wants to.

check whether this mean has changed since 2014 Suppose an economist wants to check whether this meanhas changed since 2014 The key word here is changed The mean annual earning of employed Americans haschanged since 2014 This is an example of a two tailed test .

Prem Mann Introductory Statistics 9 ECopyright 2015 John Wiley Sons All rights reserved A Two Tailed Test Let be the mean annual earning of employed Americans The two possible decisions are.

H 47 230 The mean annual earning of employedAmericans has not changed since 2014 H 47 230 The mean annual earning of employedAmericans has changed since 2014 Prem Mann Introductory Statistics 9 E.

Copyright 2015 John Wiley Sons All rights reserved A Two Tailed Test Whether a test is two tailed or one tailed is determinedby the sign in the alternative hypothesis If the alternative hypothesis has a not equal to sign it.

is a two tailed test Prem Mann Introductory Statistics 9 ECopyright 2015 John Wiley Sons All rights reserved Figure 9 2 A Two Tailed TestPrem Mann Introductory Statistics 9 E.

Copyright 2015 John Wiley Sons All rights reserved A Left Tailed TestReconsider the example of the mean amount of soda in allsoft drink cans produced by a company The company claimsthat these cans on average contain 12 ounces of soda .

However if these cans contain less than the claimed amountof soda then the company can be accused of cheating Suppose a consumer agency wants to test whether the meanamount of soda per can is less than 12 ounces Note that thekey phrase this time is less than which indicates a left tailed.

Prem Mann Introductory Statistics 9 ECopyright 2015 John Wiley Sons All rights reserved A Left Tailed Test Let be the mean amount of soda in all cans The twopossible decisions are.

H 12 ounces The mean is equal to 12 ounces H1 12 ounces The mean is less than 12 ounces Prem Mann Introductory Statistics 9 ECopyright 2015 John Wiley Sons All rights reserved A Left Tailed Test.

In this case we can also write the null hypothesis as H0 12 This will not affect the result of the test as long as thesign in H1 is less than When the alternative hypothesis has a less than sign thetest is always left tailed .

Prem Mann Introductory Statistics 9 ECopyright 2015 John Wiley Sons All rights reserved Figure 9 3 A Left Tailed TestPrem Mann Introductory Statistics 9 ECopyright 2015 John Wiley Sons All rights reserved .

A Right Tailed TestAccording to www city data com the average price of homesin West Orange New Jersey was 471 257 in 2013 Suppose a real estate researcher wants to check whetherthe current mean price of homes in this town is higher than.

471 257 The key phrase in this case is higher than whichindicates a right tailed test Prem Mann Introductory Statistics 9 ECopyright 2015 John Wiley Sons All rights reserved A Right Tailed Test.

Let be the current mean price of homes in this town Thetwo possible decisions are H 471 257 The current mean price of homes inthis town is not higher than 471 257 H 471 257 The current mean price of homes in.

this town is higher than 471 257 Prem Mann Introductory Statistics 9 ECopyright 2015 John Wiley Sons All rights reserved A Right Tailed TestWhen the alternative hypothesis has a greater than sign .

the test is always right tailed Prem Mann Introductory Statistics 9 ECopyright 2015 John Wiley Sons All rights reserved Figure 9 4 A Right Tailed TestPrem Mann Introductory Statistics 9 E.

Copyright 2015 John Wiley Sons All rights reserved Table 9 3 Signs in H0 and H1 and Tails of a TestPrem Mann Introductory Statistics 9 ECopyright 2015 John Wiley Sons All rights reserved Two Procedures.

Two procedures to make tests of hypothesis1 The p value approach2 The critical value approachPrem Mann Introductory Statistics 9 ECopyright 2015 John Wiley Sons All rights reserved .

9 2 Hypothesis Tests About KnownThree Possible CasesCase I If the following three conditions are fulfilled 1 The population standard deviation is known2 The sample size is small i e n 30 .

3 The population from which the sample is selected isnormally distributed Prem Mann Introductory Statistics 9 ECopyright 2015 John Wiley Sons All rights reserved Hypothesis Tests About Known.

Three Possible CasesCase II If the following two conditions are fulfilled 1 The population standard deviation is known2 The sample size is large i e n 30 Prem Mann Introductory Statistics 9 E.

Copyright 2015 John Wiley Sons All rights reserved Hypothesis Tests About KnownThree Possible CasesCase III If the following three conditions are fulfilled 1 The population standard deviation is known.

2 The sample size is small i e n 30 3 The population from which the sample is selected isnot normally distributed or its distribution isPrem Mann Introductory Statistics 9 ECopyright 2015 John Wiley Sons All rights reserved .

Hypothesis Tests About KnownThree Possible CasesPrem Mann Introductory Statistics 9 ECopyright 2015 John Wiley Sons All rights reserved Hypothesis Tests About Known.

DefinitionAssuming that the null hypothesis is true the p value can bedefined as the probability that a sample statistic such as thesample mean is at least as far away from the hypothesizedvalue in the direction of the alternative hypothesis as the one.

obtained from the sample data under consideration Note thatthe p value is the smallest significance level at which the nullhypothesis is rejected Prem Mann Introductory Statistics 9 ECopyright 2015 John Wiley Sons All rights reserved .

Figure 9 5 The p Value for a Right Tailed TestPrem Mann Introductory Statistics 9 ECopyright 2015 John Wiley Sons All rights reserved Figure 9 6 The p Value for a Two Tailed TestPrem Mann Introductory Statistics 9 E.

Copyright 2015 John Wiley Sons All rights reserved forCalculating the z Value When using the normal distribution the value of z for for atest of hypothesis about is computed as follows .

x z where x The value of z calculated for using this formula is also calledthe observed value of z Prem Mann Introductory Statistics 9 E.

Copyright 2015 John Wiley Sons All rights reserved Steps to Perform a Test of Hypothesis Using thep Value Approach1 State the null and alternative hypothesis 2 Select the distribution to use .

3 Calculate the p value 4 Make a decision Prem Mann Introductory Statistics 9 ECopyright 2015 John Wiley Sons All rights reserved Example 9 1.

At Canon Food Corporation it used to take an average of 90minutes for new workers to learn a food processing job Recently the company installed a new food processingmachine The supervisor at the company wants to find if themean time taken by new workers to learn the food.

processing procedure on this new machine is different from90 minutes A sample of 20 workers showed that it took onaverage 85 minutes for them to learn the food processingprocedure on the new machine It is known that the learningtimes for all new workers are normally distributed with a.

population standard deviation of 7 minutes Find the p valuefor the test that the mean learning time for the foodprocessing procedure on the new machine is different from90 minutes What will your conclusion be if 01 Prem Mann Introductory Statistics 9 E.

Copyright 2015 John Wiley Sons All rights reserved Example 9 1 SolutionStep 1 H0 90 H1 90Step 2 The population standard deviation is known thesample size is small n 30 but the population distribution.

is normal We will use the normal distribution to find the p value and make the test Prem Mann Introductory Statistics 9 ECopyright 2015 John Wiley Sons All rights reserved Example 9 1 Solution.

x 1 56524758 minutesx 85 90z 3 19 x 1 56524758p value 2 0007 0014.

Prem Mann Introductory Statistics 9 ECopyright 2015 John Wiley Sons All rights reserved Figure 9 7 The p Value for a Two Tailed TestPrem Mann Introductory Statistics 9 ECopyright 2015 John Wiley Sons All rights reserved .

Example 9 1 SolutionStep 4 Because 01 is greater than the p value of 0014 we reject the null hypothesis at this significance level Therefore we conclude that the mean time for learning thefood processing procedure on the new machine is different.

from 90 minutes Prem Mann Introductory Statistics 9 ECopyright 2015 John Wiley Sons All rights reserved Example 9 2The management of Priority Health Club claims that its.

members lose an average of 10 pounds or more within thefirst month after joining the club A consumer agency thatwanted to check this claim took a random sample of 36members of this health club and found that they lost anaverage of 9 2 pounds within the first month of membership.

with a standard deviation of 2 4 pounds Find the p value forthis test What will your decision be if 01 What if Hypothesis Tests About Âµ: ÏƒKnown. Definition. Assuming that the null hypothesis is true, the p-value can be defined as the probability that a sample statistic (such as the sample mean) is at least as far away from the hypothesized value in the direction of the alternative hypothesis as the one obtained from the sample data under consideration.

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