Confidence Intervals With z

Statistics 2126

Introduction

• Last time we talked about hypothesis testing with the z statistic

• Just substitute into the formula, look up the p, if it is < .05 we reject H0

z x

( / n )

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

• The population mean that is…

• = 108

• n = 9 =15

• z = +/- 1.96

x

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 )

1081.96(15 / 25)

1081.96(3)

1085.88

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