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We use the sample mean as our estimate of the population mean ... population mean μ falls into this interval? ... When we make a confidence interval, we're.
Typology: Summaries
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n We use the sample mean as our estimate of the population mean. n We should report some kind of ‘ confidence ’ about our estimate. Do we think it’s pretty accurate? Or not so accurate. n What sample size n do we need for a given level of confidence about our estimate. ¨ Larger n coincides with better estimate.
Example: Mean heart rate in young adults n We wish to make a statement about the mean heart rate in all young adults. We randomly sample 25 young adults and record each person’s heart rate. ¨ Population: all young adults ¨ Sample: the 25 young adults chosen for the study
n We know that won’t exactly equal , but maybe we can provide an interval around our observed such that we’re 95% confident that the interval contains. n Something like [ - cushion, + cushion]
71 72 73 74 75 76 77 x ( )
n We could report an interval like (72.0, 76.3) and say we’re 95% sure the true population mean lies in this interval. n How do we choose an appropriate ‘cushion’? (or margin of error (MOE)) n How do we decide how ‘likely’ it is that the population mean falls into this interval?
n When we make a confidence interval, we’re not 100% sure that it contains the unknown value of the parameter of interest, i.e. , but the methods we use to construct the inter- val will allow us to place a confidence level of parameter containment with our interval. X M
95% Confidence Interval (CI) for a Population Mean n The margin of error (MOE) for the 95% CI for is where s is the standard deviation of the sample (see slide 12), which is the estimate for the population standard deviation.
2 s n
95% Confidence Interval (CI) for a Population Mean n We can write this confidence interval more formally as Or more briefly as X! E < μ < X + E X ± E
95% Confidence Interval (CI) for a Population Mean The 95% CI extends a distance equal to the margin of error on either side of the sample mean.
Example: Mean heart rate in young adults n Calculating the 95% CI for population mean heart rate: and the 95% CI is: or (72.01, 76.31)
2 s n
Interpretation of the 95% Confidence Interval (CI) for a Population Mean n We are 95% confident that this interval contains the true parameter value. ¨ Note that a 95% CI always contains. In fact, it’s right at the center of every 95% CI. ¨ I might’ve missed the with this interval, but at least I’ve set it up so that’s not very likely.
