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1 Sect. 6.1: Confidence Intervals Statistical Confidence When we calculate an, say, of the population parameter, we want to indicate how accurate the is and how we are that the result is correct. We use intervals for this purpose. Eample: SAT Scores Suppose the entire population of SAT scores has mean and standard deviation. If we repeatedly sampled 500 scores from this population, then the sample mean,, follows the Normal(, ) distribution. Assume we know the value of is 100. Then, = 1) Using the rule, we know that will fall within 2 with probability. of the population mean 2) In our case, 2 =. To say that lies within 9 points of saying that is within points of. is equivalent to Combining these two facts we can state % of all samples (of size n = 500) will capture the true value of in the interval - to +. We can rewrite this interval as. Eample cont d Suppose our sample mean was 450. Then we say that we are % confident that the unknown mean score lies in the interval (, ) = (, ). 1 Sta 245 Sec 6.1 SB

2 Remember, we don t know for sure if this interval contains sampling gives correct results % of the time.! We just know that this method of Confidence Intervals This interval we just calculated is called a % interval. In general, intervals for a parameter consist of two parts: 1) An interval calculated from the data of the form estimate The conveys how accurate we believe our guess of the true parameter value is, based on the variability of the estimate. 2) A C, which gives the that the interval captures the true parameter value in repeated samples. Many types of intervals eist for various kinds of parameters. We will concentrate on intervals for the mean of a population. Intervals for a Population Mean Choose a SRS of size n from a population having unknown mean deviation. A level C interval for is and known standard Here, is the value on the standard normal curve with area C between - and. The interval is eact when the original population distribution is normal and is approimately correct for large otherwise. 2 Sta 245 Sec 6.1 SB

3 The most commonly used intervals are C 90% 95% 99% Values of for other values of C can be found using the bottom 2 rows of Table. Eample: SAT Scores Suppose we want to find an 80% confidence interval for the mean SAT score. Recall the population distribution was Normal(, = 100) and our sample mean was = 450. Behavior of Confidence Intervals What happens to the margin of error when sample size increases? Does it increase, decrease or stay the same? How does this affect the size of the resulting confidence interval? What happens to the size of the confidence interval as we decrease the confidence level C? (Hint: what happens to the value of?) How does the size of affect the margin of error? Thus we have 3 ways of reducing the size of the confidence interval: 1) 2) 3) 3 Sta 245 Sec 6.1 SB

4 Eample A randomized comparative eperiment studied the effect of diet on blood pressure. Researchers divided 54 healthy white males at random into two groups. One group received a calcium supplement and the other a placebo. Prior to the study, the seated systolic blood pressure was measured. The mean and standard deviation of the measurements for 27 members of the placebo group was reported to be = and s = 9.3. We want to get an idea of what the mean,, is for the blood pressure of the population from which the subjects were recruited. Calculate a 68 for. Q: Is it necessary to assume the population of systolic blood pressure measurements is normal? Q: What important assumption is required for the confidence interval to be valid? Choosing the Sample Size When designing a study, part of the preparation involves deciding the subjects necessary. Usually researchers will have a desired level and error they want to attain. Let m represent the error. For the sample mean recall we have m We can substitute values for m, and determine n is n and then solve for n. The resulting formula to *****Always round your answer!!***** Eample: SAT Scores Suppose we want to estimate the true mean SAT score to within 6 points with 95% confidence. What sample size do we need? 4 Sta 245 Sec 6.1 SB

5 CAUTIONS WARNINGS: How the data was collected is important!! There is no correct method for inference from data with bias of unknown size such as that produced by Non- Dropouts Non- The M.E. covers ONLY errors and does NOT compensate for. Conclusions drawn from statistical inference ( C.I. or Hyp. Test) are valid when the data is a from the population of interest. Our formulas for the M.E. are valid only for an. Beware of data collection our formulas do NOT compensate for sampling methods. Our formulas are when there are or the distribution is and the sample size is You MUST KNOW the standard deviation,. Eample From Gallup.com: "Negative assessments of the United States' involvement in Iraq continue to outweigh positive attitudes, but only by a slight margin. This finding is statistically similar to what Gallup found in June and July, but represents a slight improvement compared with early May % of Americans disapprove of the way President George W. Bush is handling Iraq,..." These results are based on telephone interviews with a randomly selected national sample of 1,017 adults, aged 18 and older, conducted Aug. 9-11, For results based on this sample, one can say with 95% confidence that the maimum error attributable to sampling and other random effects is ±3 percentage points. In addition to sampling error, question wording and practical difficulties in conducting surveys can introduce error or bias into the findings of public opinion polls. 5 Sta 245 Sec 6.1 SB

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