Department of Management, UTSC
MGEB12 Quantitative Methods in Economics II - Lecture 01
Chapter 9 - Introduction to Hypothesis Testing & Testing of Mean
• Hypothesis testing – use to determine if statement made about value of population parameter
(mean µ, proportion p, standard deviation σ) should or should not be rejected.
Example – Ambulance company
An ambulance company says they can respond to emergencies with mean time of 12min or less. You
take 40 observations and the mean time was 13.25 min. Can you conclude that the “true” mean
responds time is less than 12min? How about if mean time from observations was 12.5min?
If the true mean is ≤ 12 min what is the chance of you getting mean = 12.5min? Getting mean =
H o null hypothesis, tentative assumption about population parameter
H = alternate hypothesis, opposite of H
1. State hypothesis you want to test and the desired test significance level
2. Take observations
3. Calculate the test statistic based on the observed data
4. Determine if reject H or cannot reject H o
Manufacture claims usually given benefit of doubt and stated as null hypothesis
• Research question should be expressed as H a
• The ≤ , ≥ , = should be used in H o
• The ,≠ should be used in H
Car currently can do 24 miles/gallon of gas on average. Wondering if new fuel system allows it to go
further. State hypothesis to be tested. Example 2
Manufacture of soft drink says each bottle filled with at least 67.6 ounces. You want to know if this is
true. State hypothesis to be tested.
Type I and type II errors
• Because hypothesis test base on sample data (eg. 40 observations) and not entire population,
errors can be made
• Define test significance level = α = P[making type I error] = P[incorrectly rejectong H givon H
Typical value of for α used in practice is .1, 0.05 or 0.01
There are 2 approaches to testing:
1. Critical value approach
2. P-value approach
2. Normal Distribution Review
• Many of the hypothesis test procedures assumes that population is normally distributed.
• Following tends to be normally distributed
Height of people
o Weight of people
o Scientific measurements
o Test scores
2 1 −(x−µ)
f(x) = e 2σ 2
Normal probability density function: for -∞ ≤ x ≤ ∞
µ = mean of x
σ = standard deviation of x
• The distribution of X is bell shape.
How to calculate p(x ≤ 2) = ?
• Convert random variable x into standard normal random variable Z and then look up value in
Normal Distribution Table at back of book.
x− µ 2 − µ 2 − µ
p(x≤ 2) = p( σ ≤ σ ) = p(z ≤ σ )
then look up probability in table.
(i) X isnormal R.V. with µ= 40 and σ = 2. Compute P[ X ≤ 45], P[ X ≥ 35], P[ 35 ≤ X ≤ 45], P[ X ≥ 50]. (ii) P [ Z ≤ a ] = 0. 937, a =?
P [ Z ≤ a ] = 0.9, a = ?
P[ Z ≤ 1.53] = ? (ans=.9372) you type = NORMSDIST(1.53)
If p[ Z ≤ a ] = .99, a = ? (ans=2.326) you type =NORMSINV(.99)