6.5 How close are sample means to population means?
•sampling distribution : possible values of a statistic
•sampling distribution of ¯
X
–sample mean of ¯
Xis the same as X, we call it µ.
–standard error σ¯
X=σ/√n, where σX=σ.
•standard error is the variability we can expect from different studies on the same
topic.
•how does the sample size (n) affect the standard error of ¯
X?
•ex : let X= daily sales for pizza parlor A. we are given that µ= 900 and σ= 300.
then we have that ¯
X= avg weekly sales.
(a) what is the sample mean of weekly avg sales?
(b) how much variability do you expect from week to week (in the mean)?
•ex 19 lotto game : pay $1 to win $5 if you guess the correct number from 0 to 9. we
know that µ= 0.5 and σ= 1.5 for one game played. find the mean and standard
error of the sampling distribution of your mean winnings if you play
(a) once a week for the next year
(b) once a day for the next year
•what does it mean when the variability decreases? you are more likely to end up
closer to the mean. in the short run you expect more variation, in the long run you
expect less. (law of large numbers).
•sample mean has normal distribution for large n.
•ex :X= max of two dice
Xhas the distribution on slide.
we will roll the two die 30 times so that we have 30 values for X. then we will take
the mean ¯
X. we will calculate ¯
X10,000 times. the sampling distribution for ¯
Xis in
the next slide.
¯
Xhas the normal distribution with µ= 4.5 and σ= 1.4/√30.
this is called the central limit theorem.
•ex 20 lotto redux : what is the probability you come out ahead if you play the game
(a) once
(b) once a week for the next year
(c) once a day for the next year
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