Transcript Confidence Intervals With z

Confidence Intervals With z
Statistics 2126
Introduction
• Last time we talked about hypothesis
testing with the z statistic
z
x  
( / n )
• Just substitute into the formula, look up
the p, if it is < .05 we reject H0

Estimation
• We could also estimate the value of the
population mean
• Well all we will do in essence is use the
data we had, and the critical value of z
– The critical value is the value of z where p
= .05
– So for a two tailed hypothesis it is 1.96
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Back to the table…
• What value gives
you .025 in each
tail?
• You could look it up
in the entries in the
table, or use the
handy dandy web
tool I talked about
last time
So now with the old data from
last time let’s estimate the
mean
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The population mean that is…
x = 108
n=9
 =15
z = +/- 1.96
z( / n )  x  
  x  z( / n )
  108 1.96(15/ 9)
  108 1.96(5)
  108 9.8
98.2    117.8
Now be careful…
• That is the 95 percent confidence
interval for the estimate of 
• That does not mean that  moves
around and has a 95 percent chance of
being in that interval
• Rather, it means that there is a 95
percent chance that the interval
captures the mean
Two sides of the same coin
• You could use the confidence interval to
do the hypothesis test.
• Remember our null was that =100
• Well, the 95 percent confidence interval
captures 100 so the  of our group,
statistically, is no different than 100
Making our estimate more
accurate
• How could we make our estimate more
precise?
• Increase n
• Decrease z
– If we decrease z we get more false
positives though right
  x  z( / n )
  108 1.645(15/ 9)
  108 1.645(5)
  108 8.225
99.775   116.225
  x  z( / n )
  108 1.96(15/ 25)
  108 1.96(3)
  108 5.88
102.12    113.88
So in conclusion
• Confidence intervals allow you to test
hypotheses and make estimates
• They are affected by the critical value of
z and the sample size
• We practically can only change the
sample size