Lecture 13
The Central Limit Theorem
The Central Limit Theorem is one of the most important theorems in statistics, and is the
principal reason why normal distribution is the most important of all probability
distribution
Example
Suppose that weekly food expenditures of McMaster students are drawn from a
μ=$120∧σ=$12
distribution having
X
Suppose we take a random sample of 100 students and calculate . What is the
́
probability thatX will be between $119 and $121?
By the CLT X has an approximately normal distribution with a mean of $120 and
σ2
variance of $1.44 (Var X = )
n
́
We want p($119≤ X ≤$121) . First, convert to the standard normal,
Z1= $119−$121 =−0.83
√$1.44
$121−$119
Z2= =0.83
√$1.44
Next, obtain the probabilities from the standard normal distribution
Therefore
p($119≤ X≤$121)=p(−0.83≤ Z≤0.83)=p(−∞≤Z≤0.83)−p(−∞≤Z≤−0.83)=0.7967−0.2033=0.5934
If we took a sample size of 100, the probability is 59% that our sample mean would be
within $1 of the unknow population mean
In R we would compute this as follows:
pnorm(0.83) – pnorm(-0.83)
[1] 0.5935 The Distribution of the Difference Between Sample Means
Suppose we wish to know whether the mean of one population is equal to the mean of
another population, e.g., whether the average IQ for first year males equals that for first
year females (it doesn’t – females scare higher on average)
We don’t know the population parameter for the two populations, so we must estimate
them
The estimators (e.g., the sample average for each sample) have sampling distributions
since they are random variables
2
Population 1: μ1,σ 1n1
μ2,σ ,n2
Population 2: 2
Formally, we want to test whether μ1=μ2
We do this by examining the value of X 1−X 2 and seeing whether the difference is
‘sufficiently’ close to 0
X1∧X2́
If the samples are large, then the distributions of are normally distributed
́ ́ ́ ́
Since X1 and X2 are random variables, then X 1−X 2 is also a random
variable, and it has a probability distribution
The probability distribution of this sample statistic is known as the ‘sampling distribution’
́ ́ ́
of θ=(X1−X 2)
On average, X 1−

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