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8.2 Estimating  when standard deviation is unknown
Notation
Population mean:
Population standard deviation:
Sample mean:
Sample standard deviation:
Assumptions: _________________
____________________
NOW there are two variables to estimate
= increased variability(Reliability of our data)
= wider CONFIDENCE INTERVAL
Two cases:
Standard deviation is KNOWN
Standard deviation is UNKNOWN
Confidence Interval:
Confidence Interval:
Margin of Error:
Margin of Error:
Determining Critical Value, tc, when standard deviation is UNKNOWN
EXAMPLE: Find the critical value, tc, for a 0.99 confidence level for a t-distribution with sample size n=5
Other KEY points to know:
1- As n increases = t decreases
2- As n increases= t decreases = shorter CONFIDENCE INTERVAL
The distribution when standard deviation is unknown is slightly different that when standard deviation is known.
t – distributions are influenced by sample size and degrees of freedom (d.f = n-1)
Distribution when standard deviation is UNKNOWN
The distribution is called a t-distribution
Properties: pg 343
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Criteria for estimating population mean when standard deviation is unknown
Assumptions:
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EXAMPLE:Choose and determine…. Z distribution, t-distribution or neither
1. n=152, x  100, s  15 , population is skewed
2.
n  8, x  100, s  15 , population is normal
3.
n  8, x  100, s  15 , population is very skewed
4.
n  150, x  100, s  15 , population is skewed
EXAMPLE: Construct a confidence interval, given a sample of body temperatures. Sample size is 106, sample mean
98.20F, and we somehow know that 0.62F is the standard deviation. Use a 95% confidence interval.
EXAMPLE: Do you want to own your own candy store? With some interest in running your own business and a decent
credit rating, you can probably get a bank loan on startup costs for franchises such as Candy Factory. Startup costs for a
random sample of candy are given as
95 173 129 95 75 94 116 100 85