If a player purchases 1000 random tickets, then we apply the Central Limit Theorem to
Z
the average amount won per ticket.
1.70
Z
and
12.57 1000
0.3975
Z
. We can now determine the probability that the player gains money. In order for the player to
win money on the 1000 tickets,
Z
must exceed 2. To find 2
P Z
we calculate the z- score for
2 Z
:
2 1.7 .3
.75 .3975
.3975
Z Z
Z z
The standard normal table gives a value of .7734 for the z-score .75, and so:
2 1 .7734 .2266
P Z
.
Exercise:
Repeat the above with 10,000 tickets.
CHAPTER 6 POINT ESTIMATION OF PARAMETERS
A. Introduction
Last week you became familiar with the normal distribution. We now estimate the
parameters of a normally distributed population by analysing a sample taken from it. In this lecture we will be concentrating on the estimation of percentages and means of populations
but do note that any population parameter can be estimated from a sample.
1. Sampling
Sampling theory takes a whole lecture on its own Since any result produced from the sample can be used to estimate the corresponding result for the population it is absolutely
essential that the sample taken is as representative as possible of that population. Common
sense rightly suggests that the larger the sample the more representative it is likely to be but also the more expensive it is to take and analyse. A random sample is ideal for statistical
analysis but, for various reasons, other methods also have been devised for when this ideal is not feasible. We will not study sampling in this lecture but just give a list of the main
methods below. Simple Random Sampling
Systematic Sampling Stratified Random Sampling
Multistage Sampling Cluster Sampling
Quota Sampling
It is usually neither possible nor practical to examine every member of a population so we use the data from a sample, taken from the same population, to estimate the something
we need to know about the population itself. The sample will not provide us with the exact truth but it is the best we can do. We also use our knowledge of samples to estimate limits
within which we can expect the truth about the population to lie and state how confident we are about this estimation. In other words instead of claiming that the mean cost of buying
a small house is, say, exactly £75 000 we say that it lies between £70
000 and £80 000.
2. Types of Parameter estimates
These two types of estimate of a population parameter are referred to as:
Point estimate - one particular value; Interval estimate - an interval centred on the point estimate.
3. Point Estimates of Population Parameters
From the sample, a value is calculated which serves as a point estimate for the population parameter of interest.
a. The best estimate of the population percentage,
, is the sample percentage, p.
b. The best estimate of the unknown population mean,
, is the sample mean,
n x
x
This estimate of is often written
ˆ
and referred to as mu hat.
c. The best estimate of the unknown population standard deviation,
, is the sample standard deviation s, where:
1 n
x x
s
2
This is obtained from the x
n-1
key on the calculator.
N.B.
n x
x s
2
from x
n
is not used as it underestimates the value of
4. Interval Estimate of Population Parameter Confidence interval
