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What is a Test of Significance?
Statistical hypotheses – statements about population
parameters
Examples
Mean weight of adult males is greater than 160
Proportion of students with a 4.0 GPA is less than .01
In statistics, we test one hypothesis against another
The hypothesis that we want to prove is called the
alternative hypothesis, H a
Another hypothesis is formed that contradicts H a . This
hypothesis is called the null hypothesis, H 0
After taking the sample, we must either: Reject H 0 and
believe H a , or Fail to reject H 0 because there was not
sufficient evidence to reject it (meaning there is not
sufficient evidence to prove H a )
Types of errors
H 0 is true
H 0 is false
Fail to reject H 0
Correct
Type II error
Reject H 0
Type I error
Correct
The probability with which we are willing to risk a type I
error is called the level of significance of a test and is
denoted 
The probability of making a type II error is denoted 
The quantity 1   is known as the power of a test. It
represents the probability of rejecting H 0 when in fact
it is false
Decreasing  increases 
Sample size is the only way to control both types of error
Test Statistic – the statistic we compute to make the
decision (sampling distribution of the test statistic
must be known)
The p-value of a hypothesis test is the smallest value of 
such that H 0 would have been rejected
If p - value   , reject H 0
If p - value   , fail to reject H 0
Steps of a hypothesis test
1) State H a and H 0
2) Calculate the test statistic
3) Identify the p-value
4) Make decision and interpret results
Example
The current treatment for a type of cancer produces
remission 20% of the time. An investigator wishes to
prove that a new method is better. Suppose 26 of 100
patients go into remission using the new method.
  .05
There is not sufficient evidence to conclude the new
method is better.
Example
Do less than 50% of people prefer Murray’s Vanilla
Wafer’s when compared to other brands? Suppose that
in a taste test 42 of the 250 choose Murray’s.
  .05
Conclude with 95% confidence that less than 50% of
people prefer Murray’s Vanilla Wafer’s when compared
to other brands.
Inference about a Population Mean
Remember  x  
x 
n
is the standard deviation of the sampling
distribution which is referred to as the standard error
Z
x

n
has approximately a standard
normal distribution

Therefore, E  Z ( )
n
and the confidence interval is
xE
Example
A sample of 100 visa accounts were studied for the
amount of unpaid balance.
x  $645 and   $132
Construct a 95% confidence interval
We are 95% confident the mean unpaid balance of visa
accounts is between $619.13 and $670.87.
Construct a 99% confidence interval
We are 99% confident the mean unpaid balance of visa
accounts is between $611.00 and $679.00.
Notice that as we increase the confidence level the
interval gets wider
Example
A random sample of 500 apples yields
Assume   1.1 oz.
x  9.2 oz.
Find a 95% confidence interval
We are 95% confident the population mean weight of
apples is between 9.104 and 9.296 oz.
Example
A consumer protection agency wants to prove that
packages of Post Grape Nuts average less than 24 oz.
  .05
n  100
x  23.94
  .13
Conclude with 95% confidence that packages of Post
Grape Nuts has a mean less than 24 oz.
Example
It is desired to show the mean weight of a metal
component is greater than 4.5 oz.
  .05
n  10
x  4.59
  .504
There is not sufficient evidence to prove that the mean
weight is greater than 4.5 oz.