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CHAPTER 23CHAPTER 23

Inference for Means

A coffee machine dispenses coffee into paper cups. You’re supposed to get 10 ounces of coffee, but the amount varies slightly from cup to cup. Here are the amounts measured from a random sample of 20 cups. Is there evidence that the machine is shortchanging customers?9.9 9.7 10.0 10.1 9.99.6 9.8 9.8 10.0 9.59.7 10.1 9.9 9.6 10.29.8 10.0 9.9 9.5 9.9

I Need My Morning Coffee!!!

I Need My Morning Coffee!!! Step 1Step 1: Identify population PParameterarameter,

state the null and alternative HHypothesesypotheses, determine what you are trying to do (and determine what the question is asking). We want to know whether a particular coffee

machine is shortchanging its customers. The parameterparameter of interest is the mean amount of coffee dispensed by this machine. We assume that the mean amount is 10 ounces.

ounces10:

ounces10:0

AH

H The mean amount of coffee The mean amount of coffee dispensed is 10 oz.dispensed is 10 oz.The mean amount of coffee The mean amount of coffee dispensed is less than 10 oz.dispensed is less than 10 oz.

I Need My Morning Coffee!!! Step 2Step 2: : Verify the AAssumptionsssumptions by

checking the conditions Independence AssumptionIndependence Assumption

Randomization condition:Randomization condition: We are told that the sample was randomly selected

10% condition:10% condition: We can reasonably assume that we observed fewer than 10% of the cups of coffee dispensed by this machine.

Plausible independence condition:Plausible independence condition: There is no reason to believe that independence is violated

I Need My Morning Coffee!!! Step 2Step 2: : Verify the AAssumptionsssumptions by checking

the conditions Normality AssumptionNormality Assumption

Nearly normal condition:Nearly normal condition: Since 20 is relatively small, we need to check the sample distribution:

Depending on how you set up your window, you Depending on how you set up your window, you may get one of the following:may get one of the following:

oorr

Using Zoom Using Zoom Stat (9)Stat (9)

Adjust the Xscl Adjust the Xscl to 0.1to 0.1Both look roughly unimodal and symmetric, so it’s safe to Both look roughly unimodal and symmetric, so it’s safe to

say that the sampling distribution will be approximately say that the sampling distribution will be approximately normal.normal.

I Need My Morning Coffee!!! Step 3Step 3: If the conditions are met, NName ame the

inference procedure, state the TTest est statistic, and OObtain btain the p-value: NName the test:ame the test: We will perform a 1-sample t-test1-sample t-test

nsy

t

freedom of degrees 19)120()1(

49.320

1986.010845.9

ndf

t

t

0012. valuep

TTest est Statistics:Statistics:

OObtain the p-value:btain the p-value:

I Need My Morning Coffee!!! Step 4Step 4: MMake a decisionake a decision (reject or fail

to reject H0). SState your conclusiontate your conclusion in context of the problem using p-value: The p-value is so small, 0.0012, that we reject

the null hypothesis in favor of the alternative at the 0.05 alpha level. In other words, there is very strong evidence that the mean amount of coffee dispensed by this machine is less than the stated 10 ounces.

More Coffee, Please!!! Now that we know that this machine

is ripping us off, estimate how much it is shortchanging its customers with 95% confidence.

More Coffee, Please!!! Construct a 95% confidence interval for μ,

the mean of the population, from which the sample is drawn.Step 1: Step 1: First, state what you want First, state what you want

to know in terms of the to know in terms of the PParameterarameter and determine what the question is and determine what the question is askingasking

We wish to estimate the true meantrue mean amount, μ, of coffee that the machine is dispensing.

More Coffee, Please!!! Construct a 95% confidence interval for μ,

the mean of the population, from which the sample is drawn.Step 2: Step 2: SecondSecond, examine theexamine the

AAssumptions ssumptions and check the and check the conditionsconditions.

IndependenceIndependence: Randomization conditionRandomization condition: The cups of coffee

were randomly selected 10% condition10% condition: We safely assume that we have

less than 10% of all the coffee dispensed Plausible independence condition:Plausible independence condition: There is

no reason to believe that independence is violated

More Coffee, Please!!! Construct a 95% confidence interval for μ, the

mean of the population, from which the sample is drawn.Step 2: Step 2: SecondSecond, examine theexamine the

AAssumptions ssumptions and check the conditionsand check the conditions.NormalityNormality:

Nearly normal condition:Nearly normal condition: Since 20 is relatively small, we need to check the sample distribution:

This looks roughly unimodal and symmetric, so it’s safe to This looks roughly unimodal and symmetric, so it’s safe to say that the sampling distribution will be approximately say that the sampling distribution will be approximately normal.normal.

More Coffee, Please!!! Construct a 95% confidence interval for μ,

the mean of the population, from which the sample is drawn.Step 3Step 3: Third, Third, NName the inferenceame the inference, ,

do the work, and state the do the work, and state the IIntervalnterval..We will construct a 95% 1-sample t- 95% 1-sample t-

Interval Interval for means:

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1986.0093.2845.9

More Coffee, Please!!! Construct a 95% confidence interval for μ, the

mean of the population, from which the sample is drawn.Step 4Step 4: Fourth, last but not least, state Fourth, last but not least, state

your your CConclusiononclusion in context of the in context of the problemproblem

We are 95% confident that the machine We are 95% confident that the machine dispenses an average of between 9.75 dispenses an average of between 9.75 to 9.94 ounces of coffee per cup.to 9.94 ounces of coffee per cup.

Example As always, you can do all of Step 3 in

your calculator. Although the calculator will do Step 3, Although the calculator will do Step 3,

you still need to Steps 1, 2, and 4 on you still need to Steps 1, 2, and 4 on your own!!!your own!!!

What if we have no data?What if we have no data?We can compute a CI or HT using We can compute a CI or HT using

the sample’s mean and standard the sample’s mean and standard deviation. In other words, we can deviation. In other words, we can use Stats rather than Data in the use Stats rather than Data in the Inference function of the calculator. Inference function of the calculator. Let’s look at another example. Let’s look at another example.

Fishing For a Good Fishing Line Suppose you take an SRS of 53 Suppose you take an SRS of 53

lengths of an 85 lb. fishing line. Your lengths of an 85 lb. fishing line. Your sample has an average strength of 83 sample has an average strength of 83 lbs. with a standard deviation of 4 lbs. with a standard deviation of 4 lbs. Determine if the fishing line lbs. Determine if the fishing line should actually be considered an 85 should actually be considered an 85 lb. line.lb. line.

Step 1Step 1: Identify population PParameterarameter, state the null and alternative HHypothesesypotheses, determine what you are trying to do (and determine what the question is asking). We want to know whether a particular finishing

line should be considered an 85 lb. line. The parameterparameter of interest is the mean weight that the fishing line can hold. We assume that the mean weight that the line can hold is 85 lbs.

.85:

.85:0

lbsH

lbsH

A

The mean weight held is 85 lbs.The mean weight held is 85 lbs.

The mean weight held is not 85 lbs.The mean weight held is not 85 lbs.

Fishing For a Good Fishing Line

Step 2Step 2: : Verify the AAssumptionsssumptions by checking the conditions

Independence AssumptionIndependence Assumption Randomization condition:Randomization condition: We are told that

the sample is an SRS 10% condition:10% condition: We can reasonably assume

that we observed fewer than 10% of all lengths of 85 lb. fishing line.

Normality AssumptionNormality Assumption Nearly normal condition:Nearly normal condition: Since a sample of

53 is relatively large, we can say that the sampling distribution will be approximately normal by the CLT.

Fishing For a Good Fishing Line

Step 3Step 3: If the conditions are met, NName ame the inference procedure, state the TTest est statistic, and OObtain btain the p-value: NName the test:ame the test: We will perform a 1-sample t-test1-sample t-test

52,64.3 dft

Fishing For a Good Fishing Line

TTest est Statistics:Statistics:OObtain the p-value:btain the p-value: 000628. valuep

Step 4Step 4: MMake a decisionake a decision (reject or fail to reject H0). SState your conclusiontate your conclusion in context of the problem using p-value: The p-value is so small, .000628, that we would

rarely see such values from sampling error, so we reject the null hypothesis in favor of the alternative at the 0.05 alpha level. In other words, there is very strong evidence that the mean weight that the fishing line can use is not 85 lbs; the fishing line should not be considered an 85 lb. line.

Fishing For a Good Fishing Line

Fishing For a Good Fishing Line Suppose you take an SRS of 53 lengths of

an 85 lb. fishing line. Your sample has an average strength of 83 lbs. with a standard deviation of 4 lbs. Now make a Now make a 95% confidence interval for the mean 95% confidence interval for the mean strength of this type of fishing line.strength of this type of fishing line.

Fishing For a Good Fishing Line Step 1: Step 1: First, state what you want to First, state what you want to

know in terms of the know in terms of the PParameterarameter and and determine what the question is determine what the question is askingasking We wish to estimate the true meantrue mean strength

of a certain type of fishing line with 95% confidence; we will produce a 95% confidence interval.

Step 2: Step 2: SecondSecond, examine theexamine the AAssumptions ssumptions and check the and check the conditionsconditions. These are shown to be satisfied in the These are shown to be satisfied in the

previous problem.previous problem.

Fishing For a Good Fishing Line Step 3Step 3: Third, Third, NName the inferenceame the inference, do the , do the

work, and state the work, and state the IIntervalnterval.. We will use a We will use a 1-sample t-interval1-sample t-interval for the mean for the mean We will use the t-distribution with (n – 1) = 52 We will use the t-distribution with (n – 1) = 52

degrees of freedomdegrees of freedom

84.103) ,897.81(

Fishing For a Good Fishing Line Step 4Step 4: Fourth, last but not least, Fourth, last but not least,

state your state your CConclusiononclusion in context of in context of the problemthe problem We are 95% confident that the true mean We are 95% confident that the true mean

strength of the fishing line is between strength of the fishing line is between 81.9 and 84.1 pounds. 81.9 and 84.1 pounds.

Assignment

Chapter 23

Lesson:Inference for

Means

Read:Chapter

23

Problems:

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