This is the probability that the estimate procedure will generate an interval

This is the probability that the estimate procedure

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This is the probability that the estimate procedure will generate an interval that does not contain The confidence coefficient (1- α) is the probability that it will contain (.95 for example) Confidence level = 100(1- α) % (aka 95%) 11
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Confidence In order to figure out z-scores to use for the confidence intervals, we use the z-score at α/2 We take a confidence we want, find z by doing 1-α/2 12
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Example For a 95% interval, α/2 = .05/2 = 0.025 so I look up 1-.025 = . 975 on the table to see what’s to the left of that. I get z= 1.96 For a 90% interval, α/2 = .1/2 = .05 so I look up 1-.05 = .95 on the table and see z = 1.645 What about 99%? 13
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For all problems Figure out the z-score from the confidence level and use it in the formula: - z/ <= <= + z/ Margin of error = z/ For a given confidence level, the larger the population standard deviation, the wider the confidence level. 14
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Example Let the sd of a population be 0.05 in the last example instead of 0.03 Compute the 95% confidence interval based on the sample information. Our mean was 1.02 and sample size was 25 - z/ <= <= + z/ 15
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Example Let the sd of a population be 0.05 in the last example instead of 0.03 Compute the 95% confidence interval based on the sample information. Our mean was 1.02 and sample size was 25 - z/ <= <= + z/ 1.02 +/- 1.96(.05/ ) 1.02 +/- 0.02 = 1 to 1.04 The width is 04 here (.02 X 2) 16
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Other Rules For a given confidence level and population standard deviation, the smaller sample n, the wider the confidence interval. For a given sample size n and population standard deviation, the greater the confidence level, the wider the confidence interval. 17
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Example Compute the 99% confidence interval instead from the last example. sd =0.03, mean= 1.02, n=25 - z/ <= <= + z/ 18
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Example Compute the 99% confidence interval instead from the last example. sd =0.03, mean= 1.02, n=25 - z/ <= <= + z/ 1.02 +/- 2.576 (/ = 1.02 +/- .015 = 1.005 to 1.035 19
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Confidence interval for pop mean when sigma is unknown In order to use a confidence interval for a population mean, Xbar needs to be normally distributed. Another standardized statistic T is used which uses S as a replacement for sigma when the population standard deviation is not known. T = This is called a Student’s distribution, or a t distribution 20
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T-distribution If a random sample of size n is taken from a normal distribution with a finite variance, then the statistic T = follows the T distribution with (n-1) degrees of freedom. The T distribution is actually a family of distributions which are similar to x in that they are bell shaped and symmetric around 0. However, T distributions all have slightly broader tails. The degrees of freedom define the broadness of the tails of the distribution. The fewer the “df” the broader the tails. They are also the number of independent pieces of information that go into the calculation of a given statistic and can be “freely chosen.” 21
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T table Just like the normal table, there is a T-table on Table 2.
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