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4/9/2013
8‐1: Estimating µ When σ is Known
Point Estimate
• An estimate of a population parameter given by a single number.
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Margin of Error
• Even if we take a very large sample size, differ from µ.
will x
Margin of Error  x  
Confidence Levels
• A confidence level, c, is any value between 0 and 1 that corresponds to the area under the standard normal curve between –zc and +zc.
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Critical Values
Critical Value Example
• Use Table 5 to find z0.99 such that 99% of the area under the standard normal curve lies between –z0.99 and z0.99.
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Critical Value Example
Observe P(z < ‐ 2.58) ≈ 0.005 so use z0.99 = 2.58.
Thus, P(‐2.58 < z < 2.58) ≈ 0.99.
Common Confidence Levels
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Recall From Sampling Distributions
• If we take samples of size n from our population, then the distribution of the sample mean has the following characteristics:
Mean of x  x  x
Standard Deviation of x  x  x
n
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A Probability Statement
• In words, c is the probability that the sample mean will differ from the population mean by at most Maximal Margin of Error
• Since µ is unknown, the margin of error | ‐
µ| is unknown.
x
• Using confidence level c, we can say that differs from µ
by at most: x
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Confidence Intervals
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Confidence Interval Example
• Salaries of community college math instructors are historically known to be normally distributed with a standard deviation of $3,500. Using the telephone, you take a random sample of 33 instructors and calculate the mean salary to be $53,128. Determine a 0.95 confidence interval for the mean salary of all community college math instructors.
Confidence Interval Example n  33 x  53,128   3,500
c  0.95  zc  1.96

3,500
E  zc
 1.96
 1,194.17
n
33
xE   xE
53,128  1,194.17    53,128  1,194.17
51,933.83    54,322.17
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Confidence Interval Example
• Thus, we conclude with 95% confidence that the interval from $51,933.83 to $54,322.17 contains the mean salary for all community college math instructors.
Multiple Confidence Intervals
• If we repeated the confidence interval process by taking multiple random samples of equal size, some intervals would capture µ and some would not!
– The confidence level is the approximate proportion of intervals that will capture the population parameter.
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Multiple Confidence Intervals
Choosing Sample Sizes
• When designing statistical studies, it is good practice to decide in advance:
– The confidence level
– The maximal margin of error
• Then, we can calculate the required minimum
sample size to meet these goals.
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Sample Size for Estimating μ
• If σ is unknown, use σ from a previous study or conduct a pilot study to obtain s.
Always round n up to the next integer!!
Ex: (p. 324 #1 a, d)
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ASSIGNMENT:
p. 324‐326
#2a,d; 3a,d; 4‐7
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