{ConfidenceIntervalsforPopulationMeans}
Firstwewilldeveloptheideaforthespecificcaseof95%confidenceintervals,andthenwe
willmoveontothemoregeneralformulaandsomeadditionalissuesinpractice.
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The vertical line represents the value of the population mean. 8. Page 9. 95% of our x-bars will fall with 1.96 standard errors of the true population mean.
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{Confidence Intervals for Population Means} First will move we will on develop to the more the idea general for the formula specific and case some of 95%additional confidence issues intervals, in practice. and then we
Let’s use the results of one of our earlier simulations to explain the idea. Suppose we have a population, and we know the standard deviation of the population. If sample we are mean working using with sigma a fixed divided sample by thesize, square then we root can of calculaten. the standard error of the
Then we can use the online calculator or a normal table to find the appropriate values. We value really with only opposite need tosigns find when one ofusing these the values standard as we normal know distribution. these z‐scores will be the same
For the tables and calculators we are using here, we need to find the area to the left of each that valueof the so unknown we look zup‐scores. a probability The lower of 0.025. z‐score is easiest, with 2.5% of the area below
For to the the left upper is 97.5% z‐score, or a2.5% probability of the area of 0.975. is TO THE RIGHT of the unknown value so the area
When get z = we positive type in1.96. 0.025 we find the z‐score is negative 1.96 and when we enter 0.975, we
Using the table we need to find the closest area, in this case we have illustrated 0.975, and follow it both up and left – combining the values 1.9 with 0.06 we get 1.96 as well.
This leads to the following formula for the 95% confidence interval for a population mean whenever we feel the sampling distribution of x‐bar is reasonably normal. Currently, we are also assuming that we know the population standard deviation. This allows these results. us to easily manage the probabilities required to understand the development of
Later, is not weknown will discussbut instead the moremust bepractical estimated scenario from where the data. the population standard deviation
We can interpret this interval by saying: We are 95% confident that the mean SAT‐M score interval of (467.3,482.7)all community. college students in the researcher's state is covered by the
It is important to AND state the confidence level (95%). The math parameter score of all you community are estimating college in studentsthe words in of the the researcher’s problem, here state). it is the mean SAT
Another correct way of interpreting these results is: Based students upon in theour researcher’s data, we estimate state to the be mean 475. SATThe math95% confidencescore of all intervalcommunity suggests college this value could be as low as 467.3 to as high as 482.7. Note that the confidence interval was obtained by taking 475 ± 7.7. This error means is no more that wethan are 7.7 95% points. confident that by using the sample mean to estimate μ , our
It interval is NOT (467.3, correct 482.7)”!! to say “there Why? is a 95% chance that mu, the population mean, is in the
Once you are working with a particular sample, the interval either DOES or DOES NOT contain the parameter you just won’t know which it is!!! The probability is ONLY realized when we consider repeated sampling of x‐bar and thus does process! not pertain to OUR PARTICULAR x‐bar at the moment – the probability pertains to our
95% – for ofa giventhe intervals sample, we we obtain don’t knowin repeated whether samples we have will one contain of the the 95% true which value, work 5% willor one not of the 5% which do not. In and statistics, we do NOT we useuse thethe wordwords “confidence” “probability” or or “confident” “chance” when to reflect this interpreting long the run confidence chance interval from our data. The illustrations show one of each. A sampling distribution is drawn centered at a particular mean. population mean, mu. The vertical line represents the value of the population
http://content.bfwpub.com/webroot_pubcontent/Content/BCS_4/IPS7e/Student/Statistica l%20Applets/confidenceinterval.html What we can say, is that if we repeat this PROCESS, 95% of the time, our interval will contain simply do the not true know value which of the case parameter we have andand 5%the of phrase the time “confidence” it will not! implies In practice, this processwe of repeated sampling. The words probability or chance do not and should not be used when interpreting the results of confidence intervals in practice. The interval probability we only is speak associated of being with "95% the confident"randomness which of our translates statistic intoso that an forunderstanding a particular about the process not a statement about our particular sample. This applet also illustrates nicely how the sampling distribution gives us the needed width of or ourat least interval. estimate, Even the though width we needed don’t know to capture the true 95% center of sample of the means. distribution, we can find
Using our confidence this width interval. and centering it at our estimate, the sample mean from our data, we find
Here is a way that we can use the confidence interval: Do college the resultsstudents, of thisis lower study than provide the meanevidence SAT (^) ‐thatM score μ , the in meanthe general SAT‐M population score of community of college students in that state (which is 480)? The 95% confidence interval for μ was found to be (467.3,482.7) Note that 480, the mean SAT‐M score in the general population of college students in that state, falls inside the interval, so it is one of the plausible values for μ Thus lower we than cannot 480. claimIn fact, that the it could mean be higher SAT math than score 480 based of community upon our college confidence students interval. is
If then on thewe wouldother hand, be able the to mean claim SAT that‐math the mean score SATof the math general score population of community had collegebeen 490, students is lower than 490 since all of the plausible values of mu are less than that value. In this way, we will see that confidence intervals can be used to conduct hypothesis tests.