GREEK SYMBOLS EVERYWHERE
GREEK SYMBOLS EVERYWHERE
• Population: the total set of items that we are concerned
about
• Parameter: a measure used to summarize a population
(could include mean, median, standard deviation)
• Sample: a subset of the population (assumed to be
sampled randomly, where every object has an equally
likelihood of being selected)
• Statistic: a measure used to summarize a sample (mean,
median, SD)
CALCULATING PARAMETERS: POPULATIONS
-Mean is always the same
-Standard deviation
CALCULATING STATISTICS: SAMPLES
-Mean is always the same
The difference is that here you divide by (n-1)
• When you only divide by N, you get consistently low estimates
of the population SD sigma
• For this reason, the estimate is always made with (n-1)
INTUITIVE EXPLANATION
-we can start out with any distribution (a continuous or a discrete
distribution), and if it has a mean and a standard deviation, even if it looks
nothing like a normal distribution:
NOW, WE TAKE A SAMPLE FROM THIS DIST.
• You take a sample from this distribution (1, 2, 5, 5)
• Say the first time, you take a sample size of 4 (n=4) [a
sample of 4 samples!]
• The SAMPLE refers to the SET of 4 numbers, and
the sample size or “n" tells you how many you took
in your sample
NEXT WE AVERAGE THIS SAMPLE & PLOT IT
• If we average out 1,2,5, and 5 we get 3.25 (then plot it)
• NOW, repeat this again and again and again (i.e. increase
your number of trials) and plot the mean of every single
sample you take
• You continue to take samples, size 4, aveerage them, plot
the frequency of the averages
• Say you do this 10,000 times
• Your plot will begin to look like the normal distribution
SIMULATE THIS
-Use this
simulation: http://onlinestatbook.com/stat_sim/sampling_dist
/
THE SIMULATION AND WHAT WE NOTICE ABOUT
THE CLT
-The difference between n=5 and n=10 shows a much more normal
shape, tighter around the mean
-The mean is the same between the population and the sampling
distribution
SAMPLING DISTRIBUTION OF THE SAMPLE
MEAN
• This is the name of what we just made
• To recap, you make it using the following steps
• Take a sample size n
• Average your values
• Plot them
• Do it over and over
• Plot each one
• Watch as your plot begins to approximate the normal
curve
STANDARD ERROR OF THE MEAN
• When you take a sample, then take another sample, the means
will be different.
• When you take many samples again and again, then calculate the
mean for each sample these means you can plot this to form a
distribution (sampling dist. Of the sampling mean)
• and then you can calculate the standard deviation of the
distribution of these means
This is the standard error of the mean
THERE IS A SIMPLE WAY TO TAKE THE
STANDARD ERROR
-you don’t even need to take 10,000 samples
•
•
•
•
s.e. is the standard error of the mean
sigma is the SD of the population
n is the sample size
But we rarely know the SD of the population
• We can use the second formula above to
estimate the standard deviation
EXAMPLE 1
The average woman drinks 2 L of water when active outdoors for
a day (with a standard deviation of 0.7 L). You’re planning a trip
for 50 women and you bring 110 L of water. What is the
probability that you will run out of water?
DISTRIBUTION OF THE POPULATION
This data is an estimate of the population parameters. We are not told the
distribution, but can guess at a drawing to ground our thinking
TRANSLATE THE PROBLEM INTO PROBABILITY
What we are looking for is the probability that the average woman
drinks more than 2.2 liters of water (since we brought 110 L
divided by 50 women)
ANOTHER WAY OF SAYING THIS…
• If we were to take an infinite number of samples (n=50), what
is the probability of the those contained in the sample
drinking more than 2.2 L
• This sets us up to use the sampling distribution of the sample
mean
• We can take the sampling distribution of the sampling mean
when n=50
• Remember, the mean would remain at 2 L
CALCULATE THE STANDARD ERROR
• We already got the mean, now we need to get the standard
error
• This is the same thing as the standard deviation of the
sample mean = S/sqrt(n) or 0.7/sqrt(50)
• Standard error = 0.099 (almost 0.1) /a very narrow SD
DRAW YOUR PICTURE TO CONCEPTUALIZE
WHAT YOU JUST FOUND
NEXT STEP
• Go back to the question: (we are looking for the probability
our sample will have an average of 2.2 L)
• our distribution is the plot of all possible samples. We will
run out of water if our sample mean falls above 2.2
NEXT STEP
• We are finding the probability of the area of under the curve
highlighted in green hatching
• We can use a z table to figure out what the green area is
• When we are above 2.2 L, we are 0.2 above the mean.
• If we want that in terms of standard deviations, use the
formula for the z score:
x bar - mu / sigma or (2.2-2)/0.099=0.2/0.099=2.02
• This value of 2.2 L has the same probability of being
2.02 SD above the mean
LOOK IT UP IN THE Z TABLE
Be sure and consult
your picture so you
know exactly what your
z score is telling you
TRANSLATE THE Z SCORE INTO A PROBABILITY
FINAL ANSWER
Final answer: there is a 2.17% chance we will run out
of water (i.e. get a sample of 50 people who consume
more than the mean amount of water)
EXAMPLE 2
You sample 36 apples from your farm’s harvest of
200,000 apples. The mean weight is 112 grams
(with a 40 g SD) what is the probability that the
mean weight of all 200,000 apples is between 100
and 124 grams?
THINK ABOUT WHAT THE PROBLEM WANTS
• This is asking you to conceptualize the sampling
distribution of the sample mean
• We know that if we took a sample size 35 over
and over, a distribution would form where the
sampling mean would equal the population mean
mu, and the SD of the distribution can be found
with the formula for standard error
START OUT WITH A PICTURE
CALCULATE MEAN AND S.E.
Always start by figuring out what you can:
• The mean is 112
• The standard error is: 40/sqrt(36)=6.67
FORMULATE WHAT THE PROBLEM WANTS IN
TERMS OF PROBABILITY
• Go back to the original question: this is asking us
what the probability that the population mean is
within 12 of the sample mean (x bar)
• i.e. the sample mean is within 12 of the actual
mean
• You know you’re being asked for a confidence
interval because of the range
FIND THE Z SCORE FOR 12 ABOVE OR BELOW
THE MEAN:
• use the z score formula: x bar minus mu/ sigma
Get the z score: (112-100)/6.67=1.8
Go back to the question: This is like saying
what is the probability that our sample of 36
apples is within 1.8 standard deviations of
the mean?
DRAW IT AGAIN WITH THE NEW INFORMATION
FIND THE AREA UNDER THE HATCHING
• Use the z table
INTERPRET THE PROBABILITY FROM THE Z
TABLE
• Given this z chart shows from mean to z, you need
to double it to get 1.8 SD in either direction
=0.46407*2=0.92814
PUT IT BACK INTO PLAIN WORDS
• Put everything back in English: The probability that
the sample mean is 1.8 SD from the actual mean has
a 92.8 % chance, or
• there is a 92.82% chance that the actual
population mean is within 12 grams of our sample
(between 112 & 124)
• Also we are 92.8% confident that the mean is
between 112 and 124 g