Normal Approximation

Question 1

If a list of numbers has a “bell-shaped” histogram, we can approximate areas in histogram rectangles using the…

Answer 1

Normal curve.
* We can thus also approximate chances

Question 2

Central limit theorem

Answer 2

The “appropriate” normal curve is getting closer and closer to the histogram as n increases.

Question 3

Central limit theorem: mathematical formulation

Answer 3

At each fixed value z on the number line, the height of this histogram converges to ϕ(z) (red curve), the height of the standard normal density at z, as n→∞

Question 4

How big a sample size do we need before we can assume the Central Limit Theorem has “kicked in”?

Answer 4

• if the original box is “nice”, i.e. symmetric, takes lots of different values, then n=6 may be large enough;
• if the original box is “not nice”, highly skewed and/or only takes a few different values then we may need n>1000 before the box of sums is bell-shaped.

Question 5

standardised sums and standardised averages are exactly the same:

Answer 5

True

Question 6

When sampling withour replacement…

Answer 6

• the EV(sum) is the same when sampling with
• the SE(sum) is smaller than for sample with

Question 7

Any estimate of a parameter involves some kind of

Answer 7

error

Question 8

Estimating a population mean

Answer 8

• A “representative sample” is taken, and the parameter is estimated based on the sample.
• A very common approach is to use the sample mean as the estimate of the population mean.

Question 9

The (estimated) SE may be interpreted as the

Answer 9

“likely size of the error in estimation”

Question 10

A zero-one box

Answer 10

Contains N objects/tickets, each bearing either 0 or 1.
* The parameter of interest is p, the proportion of 1s.
* Mean of the box μ = p
* The mean square is also p
* SD is sqrt( p(1 - p) )