The Central Limit Theorem
A Review of Terminology
How many cups of coffee do you drink each week?
If we asked this question to two different five person groups, we will probably get two different sample means and two different sample standard deviations. Choosing different samples from the same population will produce different statistics.
of all possible samples is called the sampling
The Five Dice Experiment:
The Central Limit Theorem
mx = mean value of x and
s x = the standard deviation of x
Rule of thumb: n > 30 is large
Suppose that we play a slot machine such you can either double your bet or lose your bet. If there is a 45% chance of winning then the expected value for a dollar wager is
1(.45) + (-1)(.55) = -.1
We can compute the standard deviation:
So the standard deviation is
If we throw 100 silver dollars into the slot machine then we expect to average a loss of ten cents with a standard deviation of
Notice that the standard deviation is very small. This is why the casinos are assured to make money.
Now let us find the probability that the gambler does not lose any money, that is the mean is greater than or equal to 0.
We first compute the z-score. We have
0 - (-.1)
Now we go to the table to find the associated probability. We get .8438. Since we want the area to the right, we subtract from 1 to get
P(z > 1.01) = 1 - P(z < 1.01) = 1 - .8438 = .1562
There is about a 16% chance that the gambler will not lose.
Sampling Distributions for Proportions
The last example was a special case of proportions, that is Boolean data. For now on, we can use the following theorem.
The new Endeavor SUV has been recalled because 5% of the cars experience brake failure. The Tahoe dealership has sold 200 of these cars. What is the probability that fewer than 4% of the cars from Tahoe experience brake failure?
p = .05 q = .95 n = 200
Next we want to find
P(x < 8)
Using the continuity correction, we find instead
P(x < 7.5)
This is equivalent to
P(p < 7.5/200) = P(p < .0375)
We find the z-score
.0375 - .05
The table gives a probability of .2090. We can conclude that there is about a 21% chance that fewer than 4% of the cars will experience brake failure.
Control Charts for Proportions
A while back we discussed how to construct a control chart. Click here for this discussion. For proportions, we can use the same tool remembering that the Central Limit Theorem tells us how to find the mean and standard deviation.
Heavenly Ski resort conducted a study of falls on its advanced run over twelve consecutive ten minute periods. At each ten minute interval there were 40 boarders on the run. The data is shown below:
Make a P-Chart and list any out of control signals by type (I, II, III).
First we find p by dividing the total number of falls by the total number of skiers:
Now we compute the mean
Now we find two and three standard deviations above and below the mean are
.36 - (2)(.08) = .20 .36 - (3)(.08) = .04
.36 + (2)(.08) = .52 .36 + (3)(.08) = .68
Now we can use this data as before to construct a control chart and determine any out of control signals.
Notice that no nine consecutive points lie on one side of the blue line, no two of three points lie above 0.52 or below 0.20, and no points lie below 0.04 or above 0.68. Hence this data is in control.