Estimating with Confidence! date: 1/30 hw 10.1, 10.2, 10.3
statistical confidence • 25 NHS students were randomly interviewed about their sleep patterns. Their mean sleep time was 6 hours. From this SRS of n=25, what can be said about the mean sleep time u of NHS students (population) • the sample mean __ has a ___ distr. • its mean is __ • the s.d. of ___ is ____ = ____
GRAPH: the sampling distribution of mean sleep time of SRS (n=25) suppose the pop s.d. is 1 hr.
the sampling distribution of mean sleep time of SRS (n=25) • the 68-95-99.7 rule says that in about __% of all samples, __ will be within __ s.d. of ___ ___ (u) • in 95% of all samples, the unknown ulies between _____ and _____. • since our sample mean __ = 6, we say we are 95% confident that the unknown NHS mean sleep time lies between ___ and ___.
95% confidence interval for u • We got these numbers by a method that gives correct results 95% of the time. • form: estimate margin of error • estimate + (critical value)(standard error of the estimate).
calculator activity 5.4 5.5 5.6 5.7 5.8 6 6.2 6.3 6.4 6.5 6.6 6.7 5.9 6.1 • sample 25 from N(6,1) pop randNorm(6,1,25) • find sample mean (stat calc 1) • construct 95% interval for u • construct nine more confidence intervals and use Zinterval (Stat-Tests) • how many contain the true pop mean? • how many would you expect?
Confidence Intervals date 1/31 hw 10.5, 10.6, 10.7
homework • #10.3 • 95% CI for the mean male score 95% of all men have scores between __ and __. population sampling distribution 95% of all samples give an interval that contains the pop mean u
confidence intervals A confidence interval has 2 parts • an interval • a confidence level, C a 95% confidence level: C = .95 C
Critical values is the upper p critical value probability p
example: NAILS • take SRS size 5 from Pop (class) • expect nail length to be ____ • plot the data • assume s.d. of population is __mm • give a 90% CI for the mean nail length • would your result differ is nails came from only women?
larger samples give shorter intervals n = 20 n = 5
LAB: CI’s http://www.cvgs.k12.va.us/DIGSTATS/main/descriptv/d_confidence.htm
Suppose we are in an orchard which has twenty apple trees, and we place ten buckets underneath each tree in order to catch falling apples. Let's consider what happens to one tree.From the ten buckets, we find a sample mean (X-bar: total number of apples/ten buckets) of 6 apples per bucket, with a standard deviation (s) of 2.8 apples. If we want to be 95% confident that the mean number of apples per bucket in the orchard is within a range based upon our one tree, assuming the collection of apples in each bucket is normally distributed,
we get a confidence interval of: • So based upon our sample and calculations, we are 95% confident that the mean number of apples per bucket for all the buckets in the orchard is between ___ and ___.
Suppose we used the same level of confidence for all calculations and calculated a confidence interval for each tree in the orchard.
LAB: How CI’s behave date: 2/1 hw 10.9, 10.12, 10.13, 10.14
semester grades data set (semester 1 grades) plot the population pop mean =___ pop s.d.=___ describe the shape sample (SRS) 4 scores write the sample mean ___= ___ write the confidence level ___= ____ write the critical value ___=____ write the standard error SE = ____ write the margin of error, ME =____ write the 90% CI for the true pop mean is ___ _____ or ( , ) when you increase the confidence level, the ME ______, so the CI gets ______ when you increase the sample size (try n=16), the ME ____, so the CI gets _____
Confidence intervals • you choose the confidence level • we want high confidence and small margin of error • high confidence • small margin of error
the margin of error gets smaller when… • gets smaller • gets smaller • n gets larger
Confidence intervals get wider as the confidence increases: • Confidence intervals get narrower as sample size increases: n=5 C=.90 n=20 C=.95 n=80 C=.99
A guy notices a bunch of targets scattered over a barn wall, and in the center of each, in the "bulls-eye," is a bullet hole. "Wow," he says to the farmer, "that's pretty good shooting. How'd you do it?" "Oh," says the farmer, "it was easy. I painted the targets after I shot the holes."
Confidence intervals are a little like that. After we make a point estimate (a bullet hole), we are going to draw a target (an interval) around the point and state the probability that real objective is in the target area. The wider the target, the greater the probability, as you'd expect. Also, the more accurate the shooting, the greater the probability. • To avoid misleading you, we should re-phrase the joke. Suppose there's only one target painted on the barn, and all of the bullet holes are within it, even though they're not all in the center. It's still pretty good shooting, and the better the shooting, the smaller the target needs to be to encompass them all. • For the reasons we discussed in the last section, the accuracy of the shooting depends on how many samples we took and on how variable they are: • If they're all over the place, there's no reason to be very confident in their mean. • If they're concentrated in one area, there's good reason to think that the real objective is in that area.
activity 5.4 5.5 5.6 5.7 5.8 6 6.2 6.3 6.4 6.5 6.6 6.7 5.9 6.1 5.4 5.5 5.6 5.7 5.8 6 6.2 6.3 6.4 6.5 6.6 6.7 5.9 6.1 5.4 5.5 5.6 5.7 5.8 6 6.2 6.3 6.4 6.5 6.6 6.7 5.9 6.1
choosing the sample size wanted: high confidence - small margin of error question: how large a sample do I need to be within ___ of the true mean with ___ confidence?
some algebra. • A Subaru dealer wants to find out the age of their customers (for advertising purposes). They want the margin of error to be 3 years old. If they want a 90% confidence interval, how many people do they need to know about? • Hence the dealer should survey at least 52 people.
some cautions date 2-2 hw p526 10.16-10.18
estimating a normal mean • data from an SRS from pop • outliers affect CI (sample mean is sensitive) • if n is small and pop is not normal confidence level C is different. • the pop s.d. is known (or sample is large) • undercoverage, nonresponse may be present
interpreting CI • we are 95% confident that the true mean ___ lies between ___ and ___ • 95% of the population lies between __ and __ • the method gives correct results in 95% of all possible samples. • the probability is 95% that the true mean falls between ___ and ___