Transcript chapter 23 - Village Christian School

CHAPTER 23
Inference for Means
1
I Need My Morning Coffee!!!

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
9.6
9.7
9.8
9.7
9.8
10.1
10.0
10.0
9.8
9.9
9.9
10.1
10.0
9.6
9.5
9.9
9.5
10.2
9.9
I Need My Morning Coffee!!!

Step 1: Identify population Parameter, state the
null and alternative Hypotheses, 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
parameter of interest is the mean amount of coffee
dispensed by this machine. We assume that the mean
amount is 10 ounces.
H 0 :   10ounces The mean amount of coffee dispensed is 10 oz.
H A :   10ounces The mean amount of coffee dispensed is less
than 10 oz.
I Need My Morning Coffee!!!


Step 2: Verify the Assumptions by checking the
conditions
Independence Assumption
 Randomization
condition: We are told that the
sample was randomly selected
 10% condition: We can reasonably assume that we
observed fewer than 10% of the cups of coffee
dispensed by this machine.
 Plausible independence condition: There is no
reason to believe that independence is violated
I Need My Morning Coffee!!!


Step 2: Verify the Assumptions by checking the
conditions
Normality Assumption
 Nearly
normal condition: Since 20 is relatively small,
we need to check the sample distribution:
Depending on how you set up your window, you may
get one of the following:
or
Using Zoom Stat (9)
Adjust the Xscl to 0.1
Both look roughly unimodal and symmetric, so it’s safe to say that
the sampling distribution will be approximately normal.
I Need My Morning Coffee!!!

Step 3: If the conditions are met, Name the
inference procedure, state the Test statistic, and
Obtain the p-value:
 Name
the test: We will perform a 1-sample t-test
y
s
n
9.845 10
t
0.1986
20
t  3.49
Test Statistics: t 
df  (n  1)  (20  1)  19 degrees of freedom
Obtain the p-value: p  value  .0012
I Need My Morning Coffee!!!

Step 4: Make a decision (reject or fail to reject
H0). State 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: First, state what you want to know in
terms of the Parameter and determine what
the question is asking
 We wish to estimate the true 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: Second, examine the Assumptions
and check the conditions.
 Independence:
Randomization condition: The cups of coffee were
randomly selected
 10% condition: We safely assume that we have less
than 10% of all the coffee dispensed
 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: Second, examine the Assumptions
and check the conditions.
 Normality:

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 say
that the sampling distribution will be approximately normal.
More Coffee, Please!!!

Construct a 95% confidence interval for μ, the
mean of the population, from which the sample
is drawn.
 Step 3: Third, Name the inference, do the
work, and state the Interval.
 We will construct a 95% 1-sample t-Interval
for means:
 0.1986
9.845 2.093
  (9.752, 9.938)
 20 
More Coffee, Please!!!

Construct a 95% confidence interval for μ, the
mean of the population, from which the sample
is drawn.
 Step 4: Fourth, last but not least, state your
Conclusion in context of the problem
 We are 95% confident that the machine
dispenses an average of between 9.75 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, you still
need to Steps 1, 2, and 4 on your own!!!
What if we have no data?
 We can compute a CI or HT using the
sample’s mean and standard deviation. In
other words, we can use Stats rather than
Data in the Inference function of the
calculator. Let’s look at another example.
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. Determine if the fishing line should
actually be considered an 85 lb. line.
Fishing For a Good Fishing Line

Step 1: Identify population Parameter, state the
null and alternative Hypotheses, 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 parameter 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.
H 0 :   85lbs. The mean weight held is 85 lbs.
H A :   85lbs. The mean weight held is not 85 lbs.
Fishing For a Good Fishing Line

Step 2: Verify the Assumptions by checking the
conditions

Independence Assumption
 Randomization condition: We are told that the
sample is an SRS
 10% condition: We can reasonably assume that we
observed fewer than 10% of all lengths of 85 lb.
fishing line.
Normality Assumption
 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 3: If the conditions are met, Name the
inference procedure, state the Test statistic, and
Obtain the p-value:
 Name
the test: We will perform a 1-sample t-test
Test Statistics: t  3.64, df  52
Obtain the p-value: p  value .000628
Fishing For a Good Fishing Line

Step 4: Make a decision (reject or fail to reject
H0). State 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

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 95% confidence interval for
the mean strength of this type of fishing line.
Fishing For a Good Fishing Line

Step 1: First, state what you want to know in
terms of the Parameter and determine what the
question is asking
 We
wish to estimate the true mean strength of a
certain type of fishing line with 95% confidence; we
will produce a 95% confidence interval.

Step 2: Second, examine the Assumptions and
check the conditions.
 These
are shown to be satisfied in the previous
problem.
Fishing For a Good Fishing Line

Step 3: Third, Name the inference, do the work,
and state the Interval.
 We
will use a 1-sample t-interval for the mean
 We will use the t-distribution with (n – 1) = 52
degrees of freedom
(81.897, 84.103)
Fishing For a Good Fishing Line

Step 4: Fourth, last but not least, state your
Conclusion in context of the problem
 We
are 95% confident that the true mean strength of
the fishing line is between 81.9 and 84.1 pounds.
Assignment
Chapter 23
Lesson:
Inference for Means
Read:
Chapter 23
Problems:
1 - 31 (odd)