Download Confidence Intervals

Accelerated Math 3
Sampling Distribution & Confidence Intervals
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Date:______________________
-------------------------------------------------------------------------------------------------------------------------------------Statistical Inference provides methods for drawing conclusions about a population from sample data. The two
major types of statistical inference are confidence intervals and tests of significance. We will focus only on
understanding and using confidence intervals in AM3. Tests of significance are a major focus of Statistics courses.
-------------------------------------------------------------------------------------------------------------------------------------Sampling Distribution of x , “x-bar” (sample mean)
In statistics, a sampling distribution is the distribution of a given statistic based on a random sample of size n. It
may be considered as the distribution of the statistic for all possible samples of a given size. The sampling
distribution depends on the underlying distribution of the population, the statistic being considered, and the
sample size used.
For example, consider a normal population with mean μ and standard deviation . Assume we repeatedly take
samples of a given size from this population and calculate the mean, x , for each sample — this statistic is called the
sample mean. Each sample will have its own average value, and the distribution of these averages will be called the
“sampling distribution of the sample mean”.
mean (  _ _ )
x
Sampling Distribution
standard deviation (  __ )
 
__
x
 
__
x

x
n
at least 10 times the size
The formula for 
the standard deviation is valid as long as the population is
of the sample, n. As the sample size increases, the value of the statistic approaches the true value

of the population
parameter.

Example 1:
a.
Suppose that the mean SAT Math score for seniors in Georgia was 550 with a standard deviation of 50
points. Consider a simple random sample of 100 Georgia seniors who take the SAT. Describe the
distribution of the sample mean scores.
b.
What are the mean and standard deviation of this sampling distribution?
c.
Use the Empirical Rule to determine between what two scores 68% of the data falls, 95% of the data falls,
and 99.7% of the data falls.
 68% of the data is between ____ and ____ and is ____ standard deviations away form the mean
 95% of the data is between ____ and ____ and is ____ standard deviations away form the mean
 99.7% of the data is between ____ and ____ and is ____ standard deviations away form the mean
For the 95% interval, this means that in 95% of all samples of 100 students from this population, the mean score for
the sample will fall within ___ standard deviations of the true population mean or ____ points from the mean.
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-------------------------------------------------------------------------------------------------------------------------------------The Basics of Confidence Intervals
Let's start our discussion with an example.
Example 2:
You are told that a population is distributed normally with a mean, = 10, and a standard deviation,
In this case, 80% of the data fall between what two values?
= 2.
From our discussions about normal curves, you should recognize the situation to be as follows:
Where the area on each side of the mean = 0.80/2 = 0.40, and where you are asked to find x min and xmax.
You also know that this can be represented as follows using the standard normal curve:
Using your z-score probability chart* you can find the values of z:
zmin = _____
zmax = _____
*You can also find these values by using the invNorm() function on your calculator.


Ti-83/84: 2nd VARS [DISTR], ARROW DOWN to select 3:invNormal(, and then press ENTER
Ti-Nspire: Menu, 5:Probability, 5: Distribution, 3: InverseNorm, ENTER
And using the rearranged form of the z-score equation, we find that:
xmax =
xmin =
In other words, for a population distributed normally with = 10 and = 2:
80% of the data fall in the range between ____ and ____. You just figured out your first confidence interval!
In the language of confidence intervals, you could say that for the situation described above:
 You are 80% confident that any point chosen at random will fall between ____ and ____.
 Your 80% confidence interval is [____, ____]
Confidence intervals are expressed in percentages, such as the 80% confidence interval or the 95% confidence
interval. The percentage values 80% and 95% are known as the confidence level.
Above, we knew the population mean, but in practice, we often do not. So we take samples and create confidence
intervals as a method of estimating the true value of the parameter. When we find a 95% confidence interval, we
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believe with 95% confidence that the true parameter falls within our interval. However, we must accept that 5% of
all samples will give intervals which do not include the parameter.
Every confidence interval takes the same shape: estimate  margin of error.
The margin of error has two main components: the number of standard deviations from the mean (i.e. the z-score)
and the standard deviation. (Margin of error = z .)
Because we do not usually know the details of a population parameter (e.g. mean and standard deviation), we must
use estimates of these values. So our margin of error becomes m = z(estimate). Therefore, the confidence interval
becomes: estimate  margin of error  estimate  z(estimate).
Example 3:
What is the 80% confidence interval for a population for which = 36.20 and
= 12.30?
We know from the last example that for an 80% confidence level, the values of z are: z min = -1.28, zmax = 1.28
And using the rearranged form of the z-score equation again, we find that:
xmax = + 1.28 =
xmin = -1.28 =
In other words, for a population distributed normally with = 36.2 and
The 80% confidence interval is [_____, _____].
= 12.3:
What are the Common Confidence Intervals?
The z-score used in the confidence interval depends on how confident one wants to be. There are a few common
levels of confidence used in practice: 90%, 95%, and 99%.
Confidence Level
Corresponding z-score
Corresponding Interval
  z*
90%
95%
99%
Example 4:
You know the standard deviation of a population, = 2, but you don't know its average, . So, in order to estimate
the average, you take 36 samples and compute a sample mean of = 9.2. What is the 90% confidence interval for ?
First, let’s identify what z, n, and

__
x
will be:

Then, let's make sure you recognize what's going on here:
 We were told we knew  ,but not 
 So we estimated  by sampling and getting x
The 90% confidence interval is [ ____ , ____ ]



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Practice Problems
1. A random sample of 100 observations is obtained from a normally distributed population
with a standard deviation of 10. What is a 95% confidence interval for the mean of the
population if the sample mean is 40?
2. You work in a factory in which the manufacturing equipment creates engine parts of a certain
average length, , and standard deviation, = 0.100 mm. The equipment did this perfectly
for years until Buddy Rogers (a not-too-swift colleague of yours) dropped a hammer on it last
week. You're pretty sure that the standard deviation hasn't changed, but you're not sure
about the mean. So, you sample 100 parts coming off the line and find that the sample mean
is 9.200 mm.
a. What is the 99% confidence interval for your estimate of the sample mean?
b. Interpret your solution.
3. Suppose that we check for water clarity in 50 locations in Lake Tahoe and discover that
the average depth of clarity of the lake is 14 feet. Suppose that we know that the standard
deviation for the entire lake's depth is 2 feet. What can we conclude about the average
clarity of the lake with a 90% confidence level?
a. Determine a 90% confidence interval for the mean clarity of the lake.
b. Interpret your solution.
4. Suppose a student measuring the boiling temperature of a certain liquid observes the
readings (in degrees Celsius) 102.5, 101.7, 103.1, 100.9, 100.5, and 102.2 on 6 different
samples of the liquid. He calculates the sample mean to be 101.82. If he knows that the
standard deviation for this procedure is 1.2 degrees, what is the confidence interval for
the population mean at a 95% confidence level?
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