We take one sample and we calculate the mean
The Mean is the estimator of the true population
parameter
Build a confidence interval, using a specific a,
such as a = 0.05
There is a 95% chance that the true population mean
lies within an interval
We take a second sample, and we get a different mean
and confidence interval
Example – Average income for men and women
Men earn on average $40,000 per year
Women earn on average $30,000 per year
Hypothesis Test
We test whether the population parameters are the
same or different
m_{men}: Population mean for men’s
income
mm_{women}: Population mean for women’s
income
Also, these hypothesis test are equivalent
A hypothesis test has to cover all situations
This hypothesis test is invalid
We are missing the case whether women have higher
incomes than “ men
We choose a Significance Level, a
Type I Error – the error we make when we reject the
true hypothesis, H_{0}
This occurs a%
of the time
Type II Error – the error we make when we fail to
reject a false null hypothesis
Denoted by b
probability
We cannot observe this in the data
Usually researchers set a
= 0.1, 0.05, or 0.01
These a’s have
a good balance between Type I and Type II errors
Example
Choose a = 1 x
10^{6}
The Type I error becomes smaller, but the Type II
error becomes larger
You are increasing the chances of “failing to
reject” a false null hypothesis
The Power is defined as 1 – b
The Power is the probability we reject the null
hypothesis when it is false
We want 1 – b to be high
How do we increase the power?
The larger the number of observations, the more
information we have; the more power
The type of statistical test

Testing
the Hypothesis if two sample means come from the same population
Example
Assuming s is
known, thus we use the normal distribution
We know from the table

Average
Income 
Observations 
Standard
Deviation 
Men 
$40,000 
200 
$10,000 
Women 
$30,000 
150 
$5,000 
We must combine the variances
We are assuming the variances are the same
Calculate the Standard Error (SE)
The ztest
In Excel, we can calculate the pvalue for this z
statistic
The function, =normdist(value, mean, std. deviation,
cumulative)
value is the z statistic
mean is zero, because it has been standardized
std. deviation is 1, because it has been standardized
cumulative =1. We are calculating areas under the
PDF (which is actually the CDF)
The pvalue = 1.55 x 10^{34}
Two cases
If the z is positive, then Excel returns [z, z], so
subtract the pvalue from 1 to get the positive tail
If the z is negative, then Excel returns the proper
pvalue for a left side tail
We usually do a two tail test
We divide a by 2
and put this probability in each tail
The two tail test is shown below
Three ways to test a hypothesis
zstatistic
Reject the null hypothesis, if
In our case, our zvalue is 12.2 and our critical z
is 1.96
Thus, reject the H_{0} and conclude men and
women have different income levels
pvalues
Reject the null hypothesis, if
In our case, our pvalue is 1.55 x 10^{34}
while our critical probability is 0.025
Thus, reject the H_{0} and conclude men and
women have different income levels
Confidence intervals
We can construct confidence intervals the test
hypothesis
This method is shown in the next lecture
In reality, we never know the population parameter, s^{2}
Thus, when we estimate s^{2},
then we switch the distribution to a tdistribution
The analysis is the same; however, the methods to
pool variances look more complicated
We only examined twotail hypothesis test; however,
this analysis can be applied to one tail hypothesis tests

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