Topic_6 - Topic 6 Confidence intervals based on a single...

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1 Topic 6 - Confidence intervals based on a  single sample Sampling distribution of the sample mean  pages  187 - 189   Sampling distribution of the sample variance  pages  189 - 190   Confidence interval for a population mean  pages  209 - 212   Confidence interval for a population variance  pages  268 - 270   Confidence interval for a population proportion  pages  278 - 281  
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2 Confidence Intervals To use the CLT in our examples, we had to know the population  mean,  μ , and the population standard deviation,  σ . This is okay if we have a huge amount of sample data to estimate  these quantities with     and s, respectively. In most cases, the primary goal of the analysis of our sample data is to  estimate and to determine a range of likely values for these population  values. x
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3 Confidence interval for  μ  with  σ  known Suppose for a moment we know  σ  but not  μ The CLT says that     is approximately normal with a mean of  μ  and a variance  of  σ 2 / n . So,                  will be standard normal. For the standard normal distribution, let  z α  be such that  P( Z  >  z α ) =  α X μ σ - = x Z n
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4 Confidence interval for  μ  with  σ  known Safety tip…. . Alpha is the total area of the curve outside the confidence  interval.  For example, if you want a 95% confidence interval, the area  between the extremes, limits or boundaries specified is 95% and the  area outside, or alpha, is 5%. /2 /2 /2 ( ) P x Z x Z x Z n n n α σ μ + -
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5 Confidence interval for  μ  with  σ  unknown For large samples ( n  ≥ 30), we can replace  σ  with  s , so that                       is a (1- α )100% confidence interval for  μ . For small samples ( n  < 30)  from a normal population , is a (1- α )100% confidence interval for  μ . The value  t α , n -1  is the appropriate quantile from a  t  distribution with  n -1  degrees of freedom. α / 2 x z s n - / 2, 1 n x t s n t-distribution calculator
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6 Acid rain data The EPA states that any area where the average pH of rain is less than 5.6 on  average has an acid rain problem. pH values collected at Shenandoah National Park are listed below. Calculate 95%  and 99% confidence intervals for the average pH of rain in the  park.
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7 Light bulb data The lifetimes in days of 10 light bulbs of a certain variety are given below.   Give a 95% confidence interval for the expected lifetime of a light bulb of this  type. Do you trust the interval?
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This note was uploaded on 03/19/2012 for the course MATH MATH 304 taught by Professor Young during the Spring '09 term at Texas A&M.

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Topic_6 - Topic 6 Confidence intervals based on a single...

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