What can we say about the average of the draws? The ...

Chapter 23: The Accuracy of Averages

What can we say about the average of the draws?

The expected value for the average of the draws is
EVave = avebox

The standard error for the average of the draws is
SEave = SEsum
number of draws

For a large number of draws, the average of the
draws will follow the normal curve with average
EVave and standard deviation SEave.

In particular,

68% of the time the average of the draws will be
between EVave – SEave and EVave + SEave.

95% of the time the average of the draws will be
between EVave – 2 (SEave) and EVave + 2(SEave).

We can use the normal curve to find the chance
that the average of the draws is in a region of
interest. We use EVave and SEave to get standard
units.

Note:

• It does not matter what is in the box.

• The histogram for the tickets in the box does not
have to follow the normal curve.

• The average of the draws will follow the normal
curve, even if the tickets in the box are 0’s and 1’s!

• In fact, we don’t even need to know what is in the
box, we just need to know the average and the SD
of the box.

Example 1. HANES women 18-24 have an average height of 64.3”
with an SD of 2.6”. Suppose we take a random sample of 100 of these
women. What is the expected value of the average height of the
women in the sample? It’s SE?

As with the percentage, multiplying the number of
draws by some number divides the SEave by the
square root of that number.

e.g. for 100 women, SEave = .26
for 400 women, SEave = .13
for 900 women, SEave = .087

Example 2. HANES women 18-24 have an average height of 64.3”
with an SD of 2.6”. Suppose we take a random sample of 100 of these
women.
a) What’s the chance the sample average will be more than 64.5”?

b) What percentage of the women are taller than 64.5”?

Example 3. For a certain gas station, the average amount of gas sold
is 9.84 gallons, with an SD of 3.92 gallons. If we take a simple random
sample of 100 purchases from this gas station, what is the chance that
the average amount of gas for these 100 purchases is more than 10
gallons?

Example 3a. For a large population of fulltime workers, the average
income is $32,500 with an SD of $20,000.
a) If I take a simple random sample of 400 of these workers, what is the
chance the average income for my sample is more than $35,000?

b) Do I need to know the incomes follow the normal curve?
c) Can we figure out what percentage of the incomes are greater than
$35,000?

The Bootstrap
When we do not know what is in the box, we
estimate the SD of the box by the SD of the
sample.

Confidence Intervals
A 95% confidence interval for the population
average is given by

Sample average ± 2(SEave)

The confidence interval is valid if the number of
draws is large enough.

Example 4. A lake contains a large number of fish of a particular type.
A simple random sample of 300 of these fish gives an average weight
of 4.13 pounds with an SD of 2.1 pounds. Find a 95% confidence
interval for the average weight of all the fish in the lake.

Example 5. A nutrition student takes a simple random sample of 100
people from a large population and carefully monitors their caloric
intake for 1 day. The average caloric intake is 2000, with an SD of 400.
a) Find a 95% confidence interval for the average caloric intake for the
population.

b) Is your confidence interval valid if the histogram for caloric intake is
not normal?

Example 6. A university has 12,000 students. A simple random sample
of 500 students has average age 22.3 years with an SD of 4.1 years.
Find a 90% confidence interval for the average age of all students at
the university.

Reminder

Normal curve calculations, including
confidence intervals, are valid if the number
of draws is large enough.

How large is “large enough”? It depends on
the box. If the box is a long way from normal
(e.g. a box with lots of 0s and very few 1s)
then the number of draws needs to be quite
large.