Statistical Analysis and Confidence Intervals: A Comprehensive Approach to Estimating Population Parameters

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Statistical Analysis and Confidence Intervals: A Comprehensive Approach toEstimating Population ParametersWhy is a 99% confidence interval wider than a 95% confidence interval?Solution)The definition of a confidence interval is that it contains the true population mean. If I have a95% confidence interval, that means I am 95% certain that the true population mean is in theinterval. If I want to be even more certain, I have to widen the interval. If I can be less certain, Ican narrow the interval.So the widest interval will be 99%,and the narrowest would be 90%.Example:you're trying to figure out where in the city Comet Donuts is in, but you really don't know forsure. A desperately hungry person hands you a map and asks you to show him where it is. Ifsomeone forces you to be 99% accurate, are you going to draw a wide or narrow circle on themap? You can't afford to be wrong-at 99% you're saying that you'll be wrong one time out of100! So you draw a big circle.If the person asking doesn't even like donuts, they're just asking for the heck of it, you can be90% accurate, so you can take a chance and draw a small circle. You'll be wrong 10% of thetime.12.A person claims to be able to predict the outcome of flipping a coin. This person is correct16/25 times. Compute the 95% confidence interval on the proportion of times this person canpredict coin flips correctly. What conclusion can you draw about this test of his ability topredict the future?Solution)WE HAVE GIVEN THAT n = 25 and p= 16/25And we need to constructthe 95% C.I. for the proportion of times this person can predict coinsflips correctly as,ṕ± 1.96 *√(ṕq^/n)

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=.64± 1.96 *√(.64*.36/25)= .64 ± .1882So the 95% C.I. is,(0.4518,0.8282)So We Are 95 OutOf 100Attemptsare confident that the values of the samples are lies b/w(.4518,.8282)15.You take a sample of 22 from a population of test scores, and the mean of your sample is60.(a) You know the standard deviation of the population is 10. What is the 99% confidenceinterval on the populationmean?Solution)We have giventhat n = 22, sample mean =60 andσ= 10The 99% C.I.for the population mean is,= sample mean ± 2.58*σ/√n= 60 ± 2.58* 10 /√22= 60 ± 5.501So, (54.499, 65.501)(b) Now assume that you do not know the population standard deviation, but the standarddeviation in your sample is 10. What is the 99% confidence interval on the mean now?Solution)Here we have given that n =22,sample mean = 60 and S = 10Here we assume that the populationstrandeddeviation isunknown, and also n<30 so we hereuse the student t distribution as,= sample mean± tα/2,vs /√n-1Where v= n-1 degree of freedom and v= 22-1 = 21= 60± 2.831* 10/√21= 60±6.178So,(53.822, 66.178)

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