1
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:
)938.9,752.9(20
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:
1 - 31 (odd)