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Chapter
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Confidence Intervals
Confidence
Intervals
Population
Mean
Population
Proportion
Section 8.4
σ Known
σ Unknown
Section 8.2
Section 8.3
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8.2 Calculating Confidence Intervals for the Mean when the Standard Deviation (σ) of a Population is Known
A confidence interval for the mean
interval (or range) around the sample mean ( ) within which true (population) mean (µ) is expected to be
µ = ± 1.96 * 15 =
A confidence level (dependent on Z(α))
probability that the interval estimate will include the population parameter of interest (1-α)
Here α = .05 => 95% confidence
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4. Confidence Intervals for the Mean, σ Known
Assumptions for section 8.2:
The sample size is at least 30 (n ≥ 30)
The population standard deviation (σ) is known
Recall from Chapter 7:
Formula for the Standard Error of the Mean
where
σ = Population standard deviation
n = Sample size
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5. Confidence Intervals for the Mean, σ Known
Formulas for the Confidence Interval for the Mean (σ Known)
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6. Confidence Intervals for the Mean, σ Known
is called the critical z-score
The variable is known as the significance level
Example: if = .10, then = is the value that encloses 90% of the area under the normal distribution and leaves 5% in each tail
The total area to the left of the
right-hand boundary is
= 0.95
The total area to the left
of the left-hand
boundary is 0.05
α/2 = 0.05
α/2 = 0.05
0.90
-1.645
1.645 z
0.95
0.05
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7. Calculating the Margin of Error
The Margin of Error ME is the amount added and subtracted to the point estimate to form the confidence interval
Margin of Error
Margin of Error
Lower
Confidence
Limit
Upper
Confidence
Limit
Point Estimate
Width of confidence interval
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8. Calculating the Margin of Error
Increasing the sample size while keeping the confidence level constant will reduce the margin of error, resulting in a narrower (more precise) confidence interval
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9. Interpreting a Confidence Interval
We are 90% confident that the true mean is between and
Population mean (µ) may/may not be in this interval 90% of the sample means drawn from samples of this population will produce confidence intervals that include that population’s mean
An incorrect interpretation is that there is 90% probability that this interval contains the true population mean
(This interval either does or does not contain the true mean, there is no probability for a single interval)
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10. Interpreting a Confidence Interval
Each interval extends from to
But varies from sample to sample x1 x2
For 90% confidence, 90% of intervals constructed contain μ ; 10% do not
Confidence Intervals
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11. Changing Confidence Levels
The significance level, ,
Probability any given confidence interval will not contain the true population mean
The confidence level of an interval is the complement to the significance level, 1 ─
i.e., a 100(1 – )% confidence interval has a significance level equal to
The confidence interval gets wider if the confidence level increases (as Z(α) increases)
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12. Changing Confidence Levels
Consider a 95% confidence interval:
0.975
encloses 95% of the area under the curve, with 2.5% in each tail
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13. Changing Confidence Levels
z-scores for the most commonly used confidence levels are shown in this table:
Confidence Level:
Significance Level:
Critical
z-score:
80%
90%
95%
98%
99%
20%
10% 5% 2% 1%
z0.10 = ±1.28
z0.05 = ±1.645
z0.025 = ±1.96
z0.01 = ±2.33
z0.005 = ±2.575
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14. Using Excel to Determine Confidence Intervals for the Mean (σ Known)
Excel’s CONFIDENCE function calculates the margin of error for confidence intervals
The CONFIDENCE function has the following characteristics:
=CONFIDENCE (alpha, standard_dev, size)
where: alpha = The significance level of the confidence interval
standard_dev = The standard deviation of the population
size = Sample size
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15. Confidence Intervals for the Mean with Small Samples when σ is Known
When the sample size is less than 30 and sigma is known, the population must be normally distributed to calculate a confidence interval
With n < 30 the Central Limit Theorem cannot be applied, so we can’t say the sampling distribution will be approximately normal…
…but the sampling distribution is always normal (regardless of sample size) if the population is normally distributed
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8.3 Calculating Confidence Intervals when the Population Standard Deviation (σ) is Unknown
Population standard deviation is unknown,
Use s, the sample standard deviation, in its place
calculate the standard error (of the mean)
Formula for the Sample Standard Deviation (recall from Chapter 3):
where:
x = The sample mean
n = The sample size (number of data values)
(xi – x ) = The difference between each data value
and the sample mean
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Confidence Intervals
Confidence
Intervals
Population
Mean
Population
Proportion
Section 8.4
σ Known
σ Unknown
Section 8.2
Section 8.3
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18. Using the Student’s t-distribution
Formula for the Approximate Standard Error of the Mean
The Student’s t-distribution
used instead of normal probability distribution when the sample standard deviation, s, is used in place of the population standard deviation, σ
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19. Using the Student’s t-distribution
Formulas for the Confidence Interval for the Mean (σ Unknown)
where:
= The sample mean
= The critical t-score
= The approximate standard error of the mean
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20. Using the Student’s t-distribution
Normal distribution
t (df = 13)
t-distributions are bell-shaped and symmetric, but have ‘fatter’ tails than the normal
t (df = 5) t
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21. Using the Student’s t-distribution
The t-distribution is a continuous probability distribution with the following properties:
It is bell-shaped and symmetrical around the mean
The shape of the curve depends on the degrees of freedom (df), df = n – 1
The area under the curve is equal to 1.0
The t-distribution is flatter and wider than the normal distribution
The critical score for the t-distribution is greater than the critical z-score for the same confidence level
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22. Using the Student’s t-distribution
Comparing t-scores and z-scores:
Confidence t t t z
Level (10 df ) (20 df ) (30 df )
Note: t z as n increases
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23. Using the Student’s t-distribution
The t-distribution is actually a family of distributions. As the number of degrees of freedom increases, the shape of the t-distribution becomes similar to the normal distribution
With more than 100 degrees of freedom (a sample size of more than 100), the two distributions are practically identical
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Using Excel and PHStat2 to Determine Confidence Intervals for the Mean (σ Unknown)
The critical t-score can be found with Excel’s TINV function, which has the following characteristics:
=TINV (alpha, degrees of freedom)
where:
alpha ( ) = The significance level of the confidence interval
degrees of freedom = n - 1
n = Sample size
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25. Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall
Confidence Intervals
Confidence
Intervals
Population
Mean
Population
Proportion
Section 8.4
σ Known
σ Unknown
Section 8.2
Section 8.3
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26. 8.4 Calculating Confidence Intervals for Proportions
Proportion data follow the binomial distribution, which can be approximated by the normal distribution under the following conditions:
nπ ≥ 5 and n(1 – π) ≥ 5
where:
π = The probability of a success in the population
n = The sample size
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27. Calculating Confidence Intervals for Proportions
The confidence interval for the proportion is an interval estimate around a sample proportion that provides us with a range of where the true population proportion lies
Formula for the Sample Proportion:
Formula for the Standard Error of the Proportion:
where:
π = The population proportion
x = The number of observations of interest in the sample (successes)
n = The sample size
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28. Calculating Confidence Intervals for Proportions
Since the population proportion π is unknown, it is estimated using the sample proportion, p
Formula for the Approximate Standard Error of the Proportion:
where:
p = The sample proportion
n = The sample size
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29. Calculating Confidence Intervals for Proportions
Formulas for the Confidence Interval for a Proportion:
where:
p = The sample proportion
= The critical z-score
= The approximate standard error of the proportion
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30. Calculating Confidence Intervals for Proportions
Formula for the Margin of Error for a Confidence Interval for the Proportion
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31. Calculating Confidence Intervals for Proportions
Example: From a random sample of U.S. citizens, 22 of 100 people are found to have blue eyes.
Calculate a 98% confidence interval for the
population proportion of blue eyes for U.S. citizens
Calculate the sample proportion and the approximate standard error
of the proportion:
Find for 98%:
3. Calculate the interval endpoints: (next slide)
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32. 8.5 Determining the Sample Size
Increasing the sample size, holding all else constant, reduces the margin of error and provides a narrower confidence interval
The sample size needed to achieve a specific margin of error can be calculated, given the following information:
The confidence level
The population standard deviation
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33. Calculating the Sample Size to Estimate a Population Mean
Solving for n:
Formula for the Sample Size Needed to Estimate a Population Mean:
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34. Calculating the Sample Size Needed to Estimate a Population Proportion
Solving for n:
Formula for the Sample Size Needed to Estimate a Population Proportion:
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35. Calculating the Sample Size Needed to Estimate a Population Proportion
In order to calculate the required sample size to estimate π, the population proportion, we need to know p, the sample proportion
Select a pilot sample and use the sample proportion, p
If it is not possible to select a pilot sample,
set p = 0.5
Setting it equal to 0.50 provides the most conservative estimate for a sample size (a sample size that is at least large enough to satisfy the margin-of-error requirement)
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36. Calculating the Sample Size Needed to Estimate a Population Proportion
Example: What sample size is needed to estimate with 95% confidence the population proportion of U.S. citizens with blue eyes within a margin of error of ± 5%? Assume a pilot sample of 100 people found 22 with blue eyes.
Find for 95%:
Calculate the required sample size:
so use a sample of size n = 264 (always round up)
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37. Calculating Confidence Intervals for Finite Populations
Formulas for the Confidence Interval for a Proportion of a Finite Population
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