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Confidence Interval for a Population
Mean:
Student’s t-Statistic
Small Sample with σ known
Use the standard normal statistic
z=
x−µ
σx
x−µ
=
σ n
Solution to Problem 2
(Small Sample with σ Unknown)
Instead of using the standard normal statistic
z=
use the t–statistic
x−µ
σx
x−µ
=
σ n
x−µ
t=
s n
in which the sample standard deviation, s, replaces the
population standard deviation, σ.
Conditions Required for a Valid SmallSample Confidence Interval for µ
• A random sample is selected from the target
population
• The population has a relative frequency
distribution that is approximately normal.
Student’s t-Statistic
The t-statistic has a sampling distribution very much like
that of the z-statistic: mound-shaped, symmetric, with
mean 0.
The primary difference
between the sampling
distributions of t and z
is that the t-statistic is
more variable than the
z-statistic.
Degrees of Freedom
The actual amount of variability in the sampling
distribution of t depends on the sample size n. A
convenient way of expressing this dependence is to say
that the t-statistic has (n – 1) degrees of freedom (df).
Student’s t Distribution
Standard
Normal
Bell-Shaped
Symmetric
‘Fatter’ Tails
t (df = 13)
t (df = 5)
0
The smaller the degrees of freedom for t-statistic, the more variable
will be its sampling distribution.
z
t
1)
2)
• We have a random sample of 15 cars of the same
model. Assume that the gas milage for the
population is normally distributed with a
standard deviaition of 5.2 miles per galon.
• A) Identify the bounds for a 90% confidence
interval for the mean given a sample mean of
22.8 miles per gallon.
• B) The car manufacturer of this particular model
claims that the average gas milage is 26 miles per
gallon. Discuss the validity of this claim using the
90% confidence interval calculated in A.
Exercise 23.18
• In 1882 Michelson measured the speed of
light. His values in km/sec and 299,000
substracted from them. He reported the
results of 23 trials with a mean of 756.22 and
a standard deviaition of 107.12.
• Find a 95% confidence interval for the true
speed of light from these statistics.
• Interpret your result.
Thinking Challenge
• We have a random sample of customer order totals
with an average of $78.25 and a population standard
deviation of $22.5.
• A) Calculate a 90% confidence interval for the mean
given a sample size of 40 orders.
• B) Calculate a 90% confidence interval for the mean
given a sample size of 75 orders.
• C) Explain the difference in the 90% confidence
intervals calculated in A and B.
• D) Calculate the minimum sample size needed to
identify a 90% confidence interval for the mean
assuming a $5 margin of error.
What’s a Statistical Hypothesis?
A statistical hypothesis is
a statement about the
numerical value of a
population parameter.
I believe the mean GPA of
this class is 3.5!
© 1984-1994 T/Maker Co.
Hypothesis Testing for Population Mean
Population
☺
☺
☺ ☺
☺
☺
I believe the
population mean
age is 50
(hypothesis).
☺
Random
sample
Mean ☺
☺X = 20
Reject
hypothesis!
Not close.
Null Hypothesis
• The null hypothesis, denoted H0, represents the
hypothesis that will be “retained” unless the data provide
convincing evidence that it is false. This usually represents
the “status quo” or some claim about the population
parameter that the researcher wants to test.
• You may think of null hypothesis as the “favored”
hypothesis; we reject it in favor of the alternative
hypothesis Ha if and only if the evidence provided by the
sample data are strong against H0 and in favor of Ha .
• “retain H0” is commonly referred to as “do not reject”.
• Stated in one of the following forms
H0: µ = (some
value) Book uses this version…
(
H0: µ ≤ (some
value)
(
H0: µ ≥ (some
value)
(
Alternative Hypothesis
1. Opposite of null hypothesis
2. The hypothesis that will be accepted only if
the data provide convincing evidence of its
truth
3. Designated Ha
4. Stated in one of the following forms
Ha: µ ≠ (some
value)
(
Ha: µ < (some
value)
(
Ha: µ > (some
value)
(
Identifying Hypotheses
Example 1: If the hypothesis of a researcher is that
the population mean is not 3, set-up the hypotheses
to be tested.
Steps:
• State the question statistically
µ≠3
• State the opposite statistically
µ=3
• State the null hypothesis statistically
H0: µ = 3
• State the alternative hypothesis statistically
Ha: µ ≠ 3
Identifying Hypotheses
Example 2: If the hypothesis of a researcher is that
the population mean is greater than 3, set-up the
hypotheses to be tested.
Steps:
• State the question statistically
µ>3
• State the opposite statistically
µ≤3
• State the null hypothesis statistically
H0: µ ≤ 3 or µ = 3
• State the alternative hypothesis statistically
Ha: µ > 3
Identifying Hypotheses
Example 3: Is the population average amount of TV
viewing 12 hours?
• State the question statistically: µ = 12
• State the opposite statistically: µ ≠ 12
• Select the alternative hypothesis: Ha: µ ≠ 12
• State the null hypothesis: H0: µ = 12
Identifying Hypotheses
Example 4: Is the population average amount of TV
viewing different from 12 hours?
• State the question statistically: µ ≠ 12
• State the opposite statistically: µ = 12
• Select the alternative hypothesis: Ha: µ ≠ 12
• State the null hypothesis: H0: µ = 12
Identifying Hypotheses
Example 5: Is the average cost per hat less than or
equal to $20?
• State the question statistically: µ ≤ 20
• State the opposite statistically: µ > 20
• Select the alternative hypothesis: Ha: µ > 20
• State the null hypothesis: H0: µ ≤ 20 or
H0: µ = 20
Identifying Hypotheses
Example 6: Is the average amount spent in the
bookstore greater than $25?
• State the question statistically: µ > 25
• State the opposite statistically: µ ≤ 25
• Select the alternative hypothesis: Ha: µ > 25
• State the null hypothesis: H0: µ ≤ 25
H0: µ = 25
Test Statistic
The test statistic is a sample statistic,
computed from information provided in the
sample, that the researcher uses to decide
between the null and alternative hypotheses.
Determining test statistic
Is the population standard deviation known?
• Yes
use below statistic even if you have a small
sample (n<30).
Test statistic
Determining test statistic
Is the population standard deviation known?
• No
If n≥30
Test statistic
If n < 30
Test statistic
If the population
has a normal
distribution
Type I Error
• A Type I error occurs if the researcher rejects
the null hypothesis in favor of the alternative
hypothesis when, in fact, H0 is true.
• The probability of committing a Type I error is
denoted by α.
• It is also called “level of significance”
Type II Error
A Type II error occurs if the researcher retains
the null hypothesis when, in fact, H0 is false.
The probability of committing a Type II error is
denoted by β.
Conclusions and Consequences for
a Test of Hypothesis
True State of Nature
Conclusion
H0 True
Ha True
Do not reject H0
Correct decision Type II error
(Assume H0 True)
(probability β)
Type I error
Reject H0
(Assume Ha True) (probability α)
Correct decision
• How will we decide if we reject the
null hypothesis?
Example
• Lets assume we would like to test
H0: µ ≤ 2400 against Ha: µ < 2400
• We have a large sample.
• By Central limit theorem sample mean will follow
an approximately normal distribution.
• So we will reject the null hypothesis if our sample
mean takes a value which is far below 2400.
• Lets assume sample mean=2000
Basic Idea
Sampling Distribution
It is unlikely
that we
would get a
sample
mean of this
value ...
If P(sample mean <2000)
is very small, then we
reject Η0 :µ = 2400.
... if in fact this were
the population mean
2000
Area= P(sample mean <2000)
µ = 2400
H0
Sample Means
Basic Idea
Sampling Distribution for z-statistics
If P(Z <z) is very small,
It is unlikely
then we reject Η0 :µ =
that we
would get a
2400.
sample
mean of this
value ...
... if in fact this were
the population mean
test statistic- z
Area= P( Z <z)=p-value
µ=0
H0
Sample Means
p-Value
•
•
•
•
•
Probability of obtaining a test statistic more
extreme (≤ or ≥) than actual sample value,
given H0 is true
Can be thought of as a measure of the
“credibility” of the null hypothesisH0 .
α is the nominal level of significance. This value
is assumed by an analyst.
p-value is also probability for making type-I
error.
But, p-value is called “observed level of
significance”.
• If p-value ≥ α, do not reject H0
• If p-value < α, reject H0
• The p-value shows our confidence to reject null
hypothesis.
• If this value is smaller than α, then the probability
that we will reject null hypothesis when it is true
is even smaller than the maximum tolerated error
probability.
• So we can conclude that null hypothesis is wrong
and can be rejected in favor of alternative
hypothesis.
• The smaller the p-value is, the more confident we
are with our decision to reject H0 .
Steps for Calculating the p-Value for a Test of
Hypothesis when σ is known or n≥30
1. Determine the value of the test statistic z
corresponding to the result of the sampling
experiment.
Steps for Calculating the p-Value for a Test of
Hypothesis when σ is known or n≥30
2a. If the test is one-tailed, the p-value is equal to the tail
area beyond z in the same direction as the alternative
hypothesis. Thus, if the alternative hypothesis is of the
form > , the p-value is the area to the right of, or
above, the observed z-value. Conversely, if the
alternative is of the form < , the p-value is the area to
the left of, or below, the observed z-value.
Steps for Calculating the p-Value for a Test of
Hypothesis when σ is known or n≥30
2b. If the test is two-tailed, the p-value is equal to twice
the tail area beyond the observed z-value in the
direction of the sign of z – that is, if z is positive, the
p-value is twice the area to the right of, or above,
the observed z-value. Conversely, if z is negative,
the p-value is twice the area to the left of, or below,
the observed z-value.
Reporting Test Results as
p-Values: How to Decide Whether to
Reject H0
1. Choose the maximum value of α that you are
willing to tolerate.
2. If the observed significance level (p-value) of the
test is less than the chosen value of α, reject the
null hypothesis. Otherwise, do not reject the
null hypothesis.
3. Typical values for α are 0.01, 0.05, 0.10.
Two-Tailed z Test
p-Value Example
Does an average box of cereal
contain 368 grams of cereal? A
random sample of 25 boxes
showed x = 372.5. The
company has specified σ to be
15 grams. Find the p-value.
How does it compare to α =
.05?
368 gm.
Two-Tailed z Test
p-Value Solution
z=
H0 : µ=368
Ha: µ≠ 368
x−µ
σ
372.5 − 368
=
= +1.50
15
25
n
0
1.50
z
z value of sample
statistic (observed)
Two-Tailed Z Test
p-Value Solution
p-Value is P(z ≤ –1.50 or z ≥ 1.50)
1/2 p-Value
1/2 p-Value
.4332
–1.50
0
From z table:
lookup 1.50
1.50
.5000
– .4332
.0668
z
z value of sample
statistic (observed)
Two-Tailed z Test
p-Value Solution
p-Value is P(z ≤ –1.50 or z ≥ 1.50) = .1336
1/2 p-Value
.0668
–1.50
1/2 p-Value
.0668
0
1.50
p-Value = .1336 ≥ α = .05
Do not reject H0.
z
One-Tailed z Test
p-Value Example
Does an average box of cereal
contain more than 368 grams
of cereal? A random sample
of 25 boxes showed x = 372.5.
The company has specified σ
to be 15 grams. Find the pvalue. How does it compare
to α = .05?
368 gm.
One-Tailed z Test
p-Value Solution
p-Value is P(z ≥1.50)
H0 : µ=368
Ha: µ> 368
Use
alternative
hypothesis
to find
direction
p-Value
.4332
0
From z table:
lookup 1.50
1.50
.5000
– .4332
.0668
z
z value of sample
statistic
One-Tailed z Test
p-Value Solution
p-Value is P(z ≥ 1.50) = .0668>α
α=0.05
Do not reject H0
p-Value
.0668
Use
alternative
hypothesis
to find
direction
.4332
0
From z table:
lookup 1.50
1.50
.5000
– .4332
.0668
z
z value of sample
statistic
p-Value
Thinking Challenge
You’re an analyst for Ford. You
want to find out if the average
miles per gallon of Escorts is less
than 32 mpg. You take a sample
of 60 Escorts & compute a sample
mean of 30.7 mpg and sample
standard deviaiton of 3.8 mpg.
What is the p-value? How does it
compare to α = .01?
•
•
•
•
H0 : µ=32 mpg
Ha: µ< 32 mpg
We have a large sample so CLT applies.
Hence we will use z-statistics
p-Value
Solution*
p-Value is P(z ≤ -2.65) = .004.
p-Value < (α = .01). Reject H0.
Use
alternative
hypothesis
to find
direction
p-Value
.004
.5000
– .4960
.0040
.4960
–2.65 0
z value of sample
statistic
z
From z table:
lookup 2.65
Calculating the p-Value for a Test of
Hypothesis when σ is unknown and
n<30
If below conditions are satisfied;
1. A random sample is selected from the target
population.
2. The population from which the sample is
selected has a distribution that is
approximately normal.
Calculating the p-Value for a Test of Hypothesis
when σ is unknown and n<30
• We use t-statistic
x−µ
t=
s n
• Lower-tailed test ( Ha:µ< µ0)
p-value=P(t n-1 < test statistic)
• Upper-tailed test ( Ha: µ> µ0 )
p-value=P(t n-1 > test statistic)
• Two-tailed test ( Ha: µ≠ µ0 )
p-value=2P(t n-1 > |test statistic|)
Example
Is the average capacity of
batteries less than 140
ampere-hours? A random
sample of 20 batteries had a
mean of 138.47 and a standard
deviation of 2.66. Assume a
normal distribution. Test at the
.05 level of significance.
One-Tailed t Test
Solution
•
•
•
•
H0: µ = 140
Ha: µ < 140
α = .05
df = 20 – 1 = 19
x − µ 138.47 − 140
t=
=
= −2.57
s
2.66
P(t19 < -2.57)= p-value= P(t19 > 2.57)
n
20
0.005<p-value<0.01
Reject at α = .05
-2.57 0
t
There is evidence population
average is less than 140
Thinking Challenge
Does an average box of
cereal contain 368 grams of
cereal? A random sample
of 25 boxes had a mean of
372.5 and a standard
deviation of 12 grams. Test
at the .05 level of
significance.
368 gm.
Thinking Challenge
You work for the FTC. A manufacturer
of detergent claims that the mean
weight of detergent is 3.25 lb. You
take a random sample of 16
containers. You calculate the sample
average to be 3.238 lb. with a
standard deviation of .117 lb. At the
.01 level of significance, is the
manufacturer correct?
3.25 lb.