Transcript Confidence Interval for Population Mean

Confidence Interval for
Population Mean
The case when the population
standard deviation is unknown
(the more common case).
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Let’s review.
When we do not know the population mean we want to use a
sample to get a feel for what the population mean might be. From
the sample we calculate a sample mean. Since we know in theory
that different samples would provide potentially different sample
means, we take our one sample mean and build a margin of error
around the sample mean. Then we have a level of confidence that
the unknown population mean is in the interval we calculated
based on the sample.
Up to know we have looked at the case where the population
standard deviation was known.
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More review
The margin of error I write about on the previous screen is
calculated using a value of Z and the standard error of the
sampling distribution.
The values of Z most commonly used are
Z
Confidence interval
1.96
95%
1.645
90%
2.58
99%
The standard error of the sampling distribution is the population
standard deviation divided by the square root of the sample
size.
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So, the margin of error is
Z times the standard error.
The confidence interval is then
(sample mean minus margin of error, sample mean plus margin of error).
New information
When the population standard deviation is not known then we
have to modify our work just a little. The standard error will still
be calculated like above. But we will not use a Z value in the
margin of error. We will use a t value.
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It turns out that when the population standard deviation is not
known the sample mean has a t distribution. The t distribution is a
lot like the normal distribution, but when we use the t distribution
we have to be aware of something called
degrees of freedom (df).
The main point for us here is that degrees of freedom = sample size
minus 1, or df = n – 1.
So, if n = 19, df = 18, if n = 11, df = 10, and so on.
Now if we want a 95% confidence interval in this case we
1) Calculate sample mean, 2) Calculate sample standard deviation,
3) Calculate standard error as sample standard deviation divided by
the square root of the sample size, 4) find our t value in the t-table
under the .025 column in the df row = n-l, 5) calculate our margin
of error as t times standard error, 6) calculate interval as sample
mean minus and plus margin of error.
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Note we look in the .025 column on the t distribution for a 95%
confidence because we would have .025 or 2.5% in the tails of the
distribution.
If we want a 99% confidence interval we look in the .005 column
and if we want a 90% confidence interval we look in the .05
column for similar reasons.
Let’s do an example.
We have the following sample values from a population: 10, 8, 12,
15, 13, 11, 6, and 5.
On the following slide I have a basic Excel printout to calculate the
sample mean and sample standard deviation.
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x
10
sample
mean =
10
8
sample
SD =
3.464
12
15
n=
8
13
df =
7
a) The point estimate of the population
mean is the sample mean = 10.
b) The point estimate of the population
standard deviation is the sample
standard deviation and when you round
to two digits you get 3.46
c) To get the 95% confidence interval
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we need to get the standard error and the
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t statistic with a upper tail value of .025
and a df = 7. The t value is 2.3646
The standard error is 3.46/sqrt(8) = 1.22. Thus the margin of error
is (2.3646)1.22 = 2.88. The interval is thus (10 – 2.88, 10 + 2.88)
= (7.12, 12.88) and thus we are 95% confident the population
mean is in the interval 7.12 to 12.88.
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Z table and t table
If you look in the Z table at a Z = 1.96 you see the value .9750.
This means .9750 of the possible Z values have values 1.96 or
less. .9750 is a cumulative value. .025 is in the upper tail.
There is 1 standard normal distribution. But the t distribution is
really a family of distributions, where each value of the degrees
of freedom defines a new distribution. When you go to the t table
in the back of the book you see across the top the values of the
upper tail area. When you go to the upper tail area .025 you see
in the df infinity row the t value is 1.96. This means when the df
is really big the t and the z distributions are the same.
See the similarity with .05 and .005?
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Another ideaIn a confidence interval we
a
b
c
d
want to focus on the middle
of the distribution. Say the
line I have between b and c is
the sample mean and the
arrows point to the low and
the high end of the interval.
If I want a 95% confidence interval (and I had the distribution
drawn in) b would have area .95/2 and c would also have .95/2.
Area a would be .05/2 and the same would work for d. So, if I have
a t distribution (or Z) why do I look at .025 when I have a 95%
confidence interval? The answer is the table works with the upper
tail and since the upper tail is just .025 we look there knowing that
the other .025 is in the lower tail.
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Problem 10 page 248
Note when working with a t you have to pick the right column and
the right row. The row is the df and equals n – 1. The column to
look at is related to the story I had on the previous slide. In the
problem we get 1 – alpha = some decimal. From this alpha = 1
minus some decimal. On the previous slide alpha was split in half
in area a and d. We focus on d.
a. alpha = .05 alpha/2 = .05/2 = .025. df = 9, the critical t =
2.2622.
b. alpha/2 = .01/2 = .005 and df = 9 so critical t = 3.2498.
c. alpha/2 = .05/2 = .025 and df = 31 so critical t = 2.0395.
d. alpha/2 = .05/2 = .025 and df = 64 so critical t = 1.9977.
b. alpha/2 = .1/2 = .05 and df = 15 so critical t = 1.7531.
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Problem 16 page 249
a. The point estimate of the population mean is the sample mean
32. Since the population standard deviation is unknown we use
the sample standard deviation to get a standard error = 9/sqrt(50)
= 1.27.
We use the t table with df = 50 – 1 = 49. The column to look in is
.025 for a 95% confidence interval. The critical t is 2.0096. The
margin of error around 32 is 2.0096(1.27) = 2.55 so the interval is
(29.45, 34.55)
b. We can be 95% confident that the population mean turnaround
time is between 29.45 to 34.55.
c. The quality improvement project was a success because the
new interval does not include and is lower than the old population
mean of 68.
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