Transcript 13. Confidence Intervals

Confidence Intervals for
Population Means
BUSA 2100, Sections 8.1, 8.2
Estimation Process
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Point estimates are single numbers, e.g.
X-bar = $35,000.
Point estimates should be close to the true
population mean (or proportion), but are
almost never exactly equal.
So we will use interval estimates, known as
confidence intervals.
Confidence intervals are a range of numbers,
e.g. $30,000 to $40,000.
Confidence Intervals
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Two conditions for confidence intervals:
1. We want them to be narrow.
For example, the interval $15,000 to $55,000
is not a useful estimate for the mean salary of
a firm’s employees. (too wide).
But the interval $34,000 - $36,000 is useful.
2. We want them to have a high probability
(usually >= 90%) of containing the population
mean (or proportion).
Confidence Levels
The probability that a confidence interval contains the population mean or
proportion is the confidence level.
 The most common confidence levels
are 90%, 95%, and 99%.
 In a normal distribution, how many std.
deviations away from the mean do we
need to go to include 95% of the items?
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Confidence Levels, Page 2
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Confidence Levels, Page 3
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Confidence Interval Formula
for Means
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Accountant Incomes Example
Example: A random sample of 64
accountants in Georgia is selected and
their annual incomes are recorded.
 The sample mean is $55,000 and the
sample std. deviation is $4,000.
 Find a 90% confidence interval for the
population mean annual income.
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Incomes Example, Page 2
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Incomes Example, Page 3
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Incomes Example, Page 4
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Part b : Find a 95% confidence interval.
Incomes Example, Page 5
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Part c: Find a 99% confidence interval if the
sample size is 400.
Confidence Intervals for Small
Samples or Sigma Unknown
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For sample sizes < 100 or when sigma is
unknown, we use the t-distribution instead
of the normal curve table. (In some situations, the normal curve table can still be used
for sample sizes much less than 100.)
Same process, except use t-values instead of
z-values.
t-values depend on the sample size, or
degrees of freedom.
The t-Distribution
The number of degrees of freedom is 1
less than the sample size; df = n - 1 .
 The t-table is based upon areas in 1 tail.
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Sample Size and t-values
Look up t-values for various confidence
levels and sample sizes.
 For small n and df values, the t-values
are very large. This creates wide confience intervals with poor accuracy.
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Sample Size and t-values,
Page 2
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Confidence Interval Formula
for Means, Small Sample Size
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Ex.: For a sample of 15 trainees, the sample
mean training time is 26 hours & the sample
std. deviation is 4.2 hours.
Find a 99% C.I. for the population mean
training time.
Training Time Example
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