The Mighty Central Limit Theorem!Find: a. the frequency distribution (histogram, too, if you’re feelin’ saucy); b. the average () ; and c. the standard deviation ()

of the sum of your two tricky dice’s pips.

2 3 4 5 6 7 8 9 10 11 120

1

2

3

4

5

6

7

Sum of Pips

Frequency

“Typical” dice

7

2.42 (ish)

Blue (skewed left) dice

8.1666...

2.65 (ish)

Sum of pips

frequency

Yellow (skewed right) dice

4.5

2.22 (ish)

Sum of pips

frequency

Green (symmetric, wide) dice

7

2.92 (ish)

Sum of pips

frequency

Orange (bimodal, asymmetric) dice

7

2.15 (ish)

Sum of pips

frequency

Red (symmetric, skewed) dice

9.1666...

1.46 (ish)

Sum of pips

frequency

For the purposes of our next activity, your dice represent the population you’re studying...now, suppose you want to find their (and ).

You’ll never be able to know them directly the way you just did...why?

So what do you do?

(Quiz (IYL): Sampling Distributions)

Can Excel help here?

The Mighty Central Limit Theorem!Suppose you want to study a population with an

average of and a standard deviation of (neither of which you know). So, you take a random sample of size n.

The Mighty Central Limit Theorem!Suppose you want to study a population with an

average of and a standard deviation of (neither of which you know). So, you take a random sample of size n.

1. Your sample mean will have a normal sampling distribution (yay!) centered on (yay!)2. This means that your sample mean is an accurate (unbiased) estimator of . In symbols,

The Mighty Central Limit Theorem!Suppose you want to study a population with an

average of and a standard deviation of (neither of which you know). So, you take a random sample of size n.

2. However, there is still variation in your sample mean (it doesn’t exactly equal ).

x

The Mighty Central Limit Theorem!Suppose you want to study a population with an

average of and a standard deviation of (neither of which you know). So, you take a random sample of size n.

3. But, your sample standard deviation s (which many articles call the standard error of the mean, or SE) shrinks as your sample size grows (yay!)3 . In fact,

SEn

The Mighty Central Limit Theorem!Suppose you want to study a population with an

average of and a standard deviation of (neither of which you know). So, you take a random sample of size n.

4. This means that, on average, will be within a “usual” range of values:

2 2x SE x SE

The Mighty Central Limit Theorem!Suppose you want to study a population with an

average of and a standard deviation of (neither of which you know). So, you take a random sample of size n.

5. Since SE is inversely proportional to the sample size, as the sample gets larger, the precision of our interval gets better (yay!)4.

?Eh

The Mighty Central Limit Theorem!Suppose you want to study a population with an

average of and a standard deviation of (neither of which you know). So, you take a random sample of size n.

6. No need for huge samples to invoke the CLT…yay5! As n , the bells become perfect, but an accepted minimum is n = 30, although, as you saw, far fewer can be fine.

Homework for The Mighty Central Limit Theorem!