Download Section 8.2 - TopCatMath

Math 160 - Cooley
Intro to Statistics
OCC
Section 8.2 – Confidence Intervals for One Population Mean When σ Is Known
Statistical Inference – Provides methods for drawing conclusions about a population from sample data.
Confidence Interval For The Mean Of A Normal Population
Draw an SRS of size n from a Normal population having unknown mean µ and known standard deviation σ. A
level (1 – α) confidence interval for µ is
x  z
The critical value z
2

2
n
is illustrated below and found in Table II.
INTERVAL PROCEDURE #1 – The One-Mean z-Interval Procedure
Purpose: To find a confidence interval for a population mean, μ.
Assumptions
1) Simple random sample
2) Normal population or large sample (n ≥ 30)
3) σ known
Step 1 – For a confidence level (1 – α), use Table II to find z 2 .
Step 2 – The confidence interval for µ is from
x  z
where z
2

2
to x  z

2
n
n
is found in Step 1, n is the sample size, and x is computed from the sample data.
Step 3 – Interpret the confidence interval.
The confidence interval is exact for normal populations and is approximately correct for large samples from
nonnormal populations.
The One-Mean z-Interval Procedure For A Population Mean
 For small samples–say, of size less than 15–the z-interval procedure should be used only when the
variable under consideration is normally distributed or very close to being so.
 For samples of moderate size–say, between 15 and 30–the z-interval procedure can be used unless the
data contains outliers or the variable under consideration is far from being normally distributed.
 For large samples–say, of size 30 or more–the z-interval procedure can be used essentially without
restriction. However, if outliers are present and their removal is not justified, you should compare the
confidence intervals obtained with and without the outliers to see what effect the outliers have. If the
effect is substantial, use a different procedure or take another sample.
 If outliers are present but their removal is justified and results in a data set for which the z-interval
procedure is appropriate (as previously stated), the procedure can be used.
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Math 160 - Cooley
Intro to Statistics
OCC
Section 8.2 – Confidence Intervals for One Population Mean When σ Is Known
A Fundamental Principle of Data Analysis
Before performing a statistical-inference procedure, examine the sample data. If any of the conditions required
for using the procedure appear to be violated, do not apply the procedure. Instead use a different, more
appropriate procedure, or, if you are unsure of one, consult a statistician.
Confidence and Precision
For a fixed sample size, decreasing the confidence level improves the precision, and vice-versa.
Margin of Error for the Estimate of μ
The margin of error for the estimate of μ is
E  z 2 

n
Note: This is simply half the length of the confidence interval.
Margin of Error, Precision, and Sample Size
The length of a confidence interval for a population mean, μ, and therefore the precision with which x
estimates μ, is determined by the margin of error, E. For a fixed confidence level, increasing the sample size
improves the precision, and vice versa.
Sample Size For Estimating μ
The sample size required for a (1 – α)-level confidence interval for μ with a specified margin of error, E, is given
by the formula
 z 2  
n  
 E




2
rounded up to the nearest whole number.
 Exercises:
Later on in the text, the term(s),  z / 2 , – z or z , will be referred to as critical value(s).
1)
For a 90% confidence level, find both critical values.
2)
For a 95% confidence level, find both critical values.
3)
For a 99% confidence level, find both critical values.
4)
For a 97% confidence level, find both critical values.
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Math 160 - Cooley
Intro to Statistics
OCC
Section 8.2 – Confidence Intervals for One Population Mean When σ Is Known
 Exercises:
5)
6)
You measure the lifetime of a random sample of 25 tires of a certain brand. The sample mean is x = 50
months. Suppose that the lifetimes for tires of this brand follow a normal distribution, with unknown
mean µ and standard deviation σ = 5 months.
a)
Find a 95% confidence interval for µ.
b)
Determine the margin of error, E.
c)
Explain the meaning of E in this context in terms of the accuracy of the estimate.
d)
Determine the sample size required to have a margin of error of 0.1 months and a 95%
confidence level.
e)
Obtain a 95% confidence interval for the mean lifetime of this certain brand of tire if a sample
of the size determined in part (d) has a mean of 50.8 months.
The time (in number of days) until maturity of a certain variety of tomato plant is normally distributed
with mean µ and standard deviation σ = 2.4. If you select a simple random sample of four plants of this
variety and measure the time until maturity. The four times, in days are: 63, 69, 62, 66. Based on these
data, find a 99% confidence interval for µ.
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Math 160 - Cooley
Intro to Statistics
OCC
Section 8.2 – Confidence Intervals for One Population Mean When σ Is Known
Interpretation and the Meaning of Confidence Intervals for µ
Consider example #5:
You measure the lifetime of a random sample of 25 tires of a certain brand. The sample mean is
x = 50 months. Suppose that the lifetimes for tires of this brand follow a normal distribution,
with unknown mean µ and standard deviation σ = 5 kg. Find a 95% confidence interval for µ.
During class we found that the 95% confidence interval for the population mean, µ, was (48.04, 51.96).
Students and teachers sometimes refer to this by saying that “We are 95% confident that µ lies in the interval
(48.04, 51.96)”.
However, while this wording is okay for student use in the classroom, it is technically incorrect, and would not
be used by actual statisticians when interpreting the meaning of a confidence interval. Because you would never
describe yourself as being 75% happy or 50% in love with someone, you also can’t be 95% confident about
something.
In actuality, a 95% confidence interval means that the method used to obtain the confidence interval
produces an interval that encapsulates the unknown, µ, 95% of the time. To be more precise a statistician
would say that if we were to take 100 samples of size 25 from the population, and from those samples construct
100 sample means, and then construct 100 confidence intervals around the sample means at the 95% level, then
95 of the 100 confidence intervals constructed from the samples would include the population mean, while the
other 5 of the 100 would not. So, for an individual sample of size 25 from the tire population there is a 95%
chance that its corresponding confidence interval encapsulates the population mean.
When interpreting the wording of confidence intervals for µ taken at level 1 – α taken from a sample of size n,
the following phrases are equivalent:


 
 .
 We are (1 – α) % confident that µ lies in the interval  x  z 2
, x  z 2
n
n

 (1 – α) % of all samples of size n taken produce an interval that contains the population mean µ.
 The method used to obtain the confidence interval produces an interval that encapsulates the unknown,
µ, (1 – α) % of the time.
 In 100 samples of size n taken from the population, (1 – α) of then will produce confidence intervals
that contain the population mean µ.
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Math 160 - Cooley
Intro to Statistics
OCC
Section 8.2 – Confidence Intervals for One Population Mean When σ Is Known
Simulation of confidence intervals
Below is a computer simulation of 20 samples of 36 new mobile home prices each. From there, 20 confidence
intervals were constructed. For the simulation, we assumed   65 (i.e., $65 thousand) and   7.2 (i.e., $7.2
thousand). We assumed these numbers based off last year’s mean and standard deviation. In reality, we don’t
know the current  , so we are assuming last year’s value for  to illustrate a point.
As you can see for the 20 samples, each one had a different sample mean. The standard deviation of the sample
mean was calculated at a constant of 2.4. The simulation shows that 19 of the 20 confidence intervals actually
contained the population mean of 65, which is 95%. If we simulated a much greater sample size, then the
confidence would be closer to the true actually 95.44%.
So, theoretically, if we took 10000 samples of 36 new mobile home prices each and constructed confidence
intervals, then we should expect 9544 of them or 95.44% of them to contain the true population mean  .
However, this is a theoretical probability. If it is not 95.44%, then it will be very close to it. Think of this model
rolling a single die 6 times. Theoretically, we should expect the outcomes to be 1,2,3,4,5, and 6.
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Math 160 - Cooley
Intro to Statistics
OCC
Section 8.2 – Confidence Intervals for One Population Mean When σ Is Known
 Sample Test Multiple Choice Questions:
7)
When a confidence level decreases,
A) the confidence interval increases
B) the confidence interval decreases
C) the margin of error increases
D) the margin of error decreases
E) Both A and C.
F) Both A and D.
G) Both B and C.
H) Both B and D.
8)
What generally happens to the sampling error as the sample size is increased?
A) It gets smaller.
B) It gets larger
C) It gets more predictable.
D) It gets less predictable.
E) None of these.
9)
Suppose a confidence interval is (4.11 , 5.77). Then the margin of error for  is:
A) 9.88
B) 23.7147
C) 1.66
D) 0.83
E) None of these.
10)
Weights of women in one age group are normally distributed with a standard deviation of 17 lb. A
researcher wishes to estimate the mean weight of all women in this age group. Find how large a sample
must be drawn in order to be 90 percent confident that the sample mean will not differ from the
population mean by more than 3.8 lb.
A) 32
B) 33
C) 54
D) 55
E) 76
AB) 77
11)
In actuality, a 95% confidence interval means that the method used to obtain the confidence interval
produces an interval that encapsulates the unknown, µ, 95% of the time. To be more precise a
statistician would say
A) that if we were to take 100 samples of size 25 from the population, and from those
samples construct 100 sample means, and then construct 100 confidence intervals around the
sample means at the 95% level, then 95 of the 100 confidence intervals constructed from the
samples would include the population mean, while the other 5 of the 100 would not.
B) we can be 95% confident that the population mean, µ, will be contained in the confidence interval.
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