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AMS7: WEEK 5. CLASS 3
Confidence Intervals for the Population Mean
Friday May 1st, 2015
Confidence Interval for the Population
mean
• Population mean: • Sample mean: • Point estimate of the Population Mean: The sample mean
is the best point estimate of the population mean
• We will learn how to build a Confidence Interval for the
population mean. This interval will be an Interval Estimate.
• We will consider two cases:
1) is known
2) is unknown
Case 1:
• Requirements for the estimation:
• 1) If the original population is itself normally distributed the
sample means of any size will be normally distributed
• 2) If the original population is not itself normally
distributed, samples with size n>30 will have sample
means approximately normal (because of the Central
Limit Theorem)
• Note: To check normality of the sample data construct a
histogram and check whether is approximately bellshaped. You can also use a normal quantile plot
Example: Confidence interval for the
population mean
• Suppose we want to estimate the mean salary of recently
graduated college students who take a statistics course. A
sample of size n=28, with sample mean salary of $55,678
is obtained. The population is assumed normally
distributed, and is known to be $9,900. Calculate a 95%
confidence interval for the population mean = average
starting salary of college graduates who have taken a
statistics course.
Example (Cont.)
• Confidence level: 95% (95% of the times, we will have a
C.I. including the true value of ).
• Also confidence level = 1-=0.95 (as a probability)
=0.05
• Margin of Error: This is the difference between the
population mean and the sample mean
= ఈ/ଶ × Critical Value
(ߪ is known)
Standard
Deviation of
the Sample Mean
Example (Cont.)
• In this example:
ఈ/ଶ =1.96
=
ଽଽ଴଴
1.96 × =
ଶ଼
3367.01
• Confidence Interval ( known)
− < < + or
± or
( − , + )
Example (Cont.)
• In this example:
52,310.99< < 59,045.01
We are 95% confident that the interval (52,310.99,
59,045.01) does contain the true value of This means: If we take many different samples of size n
and construct a CI for each sample, 95% of them would
contain the true values of Rationale of the Confidence Interval
From the Central Limit Theorem:
has a Normal distribution with mean ௑ത = and a
•
ఙ
standard deviation ௑ത =
( and area the mean and
௡
standard deviation from the parent population which is not
necessarily normal)
• We can calculate the z Score:
=
௑തିఓ೉
ഥ
ఙ೉
ഥ
௑തିఓ
=഑
ൗ ೙
=഑
ா
ൗ ೙
− z × ఙ and use the positive and
• We can get = ௡
negative values for z results
Sample size for estimating the mean
ఈ/ଶ × =
• ఈ/ଶ : Critical z Score
• E: Desired margin of Error
• : Populationstandarddeviation
ଶ
Sec. 6.3 #26
• An economist wants to estimate the mean income for the
first year of work for college graduates who majored in
biology. How many such incomes must be found if we
want to be 95% confident that the sample mean is within
$500 of the true population mean? Assume that a
previous study has revealed that for such incomes, =
$6250.
• Confidence level: 1-=0.95; =0.05; /2=0.025
• Critical value: ఈ/ଶ =1.96
• E= $500
• = $6250.
Example (Cont.)
•=
ଵ.ଽ଺×଺ଶହ଴ ଶ
=
ହ଴଴
600.25
601 (next large whole
number)
• We would need to find 601 outcomes for college
graduates who majored in biology
Case 2:
• In this case should be estimated by s, where s is the
sample standard deviation
• Now the quantity
௑തିఓ
t= ೞ
ൗ ೙
has a t distribution with n-1 degrees of freedom
This distribution is also called the Student t distribution,
developed by William Gosset (1876-1937). He was a
Guinness Brewery employee, who used the pseudonym
Student to publish his results.
Student t Distribution
Confidence Interval for the Population
mean when ߪ is unknown
• Now we use tߙ/2 as a critical value with confidence level
(1- )
• The margin of error is:
= ఈ/ଶ × • The Confidence Interval is :
− < < + Example
• Calculate a 95% Confidence Interval for the mean IQ of
biology students when n=32 is a simple random sample,
= 117.2, = 12.1
• Confidence level: 1-=0.95; =0.05; /2=0.025
• Number of degrees of freedom: 32-1=31
• Critical value: ఈ/ଶ = 2.040
• = 2.04 × • CI:
ଵଶ.ଵ
=
ଷଶ
4.36
117.2 – 4.36 < < 117.2 + 4.36
112.84< < 121.56
t Distribution
32-1 degrees
of freedom
ߙ/2
‫ݐ‬ఈ/ଶ = 2.040