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QMS202-Business Statistics II Chapter11
Chapter11 Fundamentals of Hypothesis Testing:
One-Sample tests
Outcomes:
1. Define a hypothesis and hypothesis testing
2. Describe the hypothesis-testing procedure
3. Distinguish between a one-tailed and a two-tailed test of
hypothesis
4. Conduct a hypothesis test for a population mean with known
population standard deviation
5. Conduct a hypothesis test for a population mean with unknown
population standard deviation
6. Conduct a hypothesis test for a population proportion
7. Define Type I and Type II errors
8. Compute and interpret a p-value
What is a Hypothesis?
1. A hypothesis is a statement about a population.
2. In statistical analysis we make a claim, that is, state a hypothesis, collect data,
and then use the data to test the claim.
3. In most cases the population is so large that it is not feasible to study all the
items, objects or persons in the population. For example, it would not be possible
to contact every accountant in Canada to find out his or her annual income.
4. An alternative to measuring or interviewing the entire population is to take a
sample from the population and test a statement to determine whether the
sample does or does not support the statement concerning the population.
What is Hypothesis Testing?
1. A procedure based on sample evidence and probability theory which
determine whether the hypothesis is a reasonable statement.
2. The procedure for testing a hypothesis
Step1. Define the parameter(s) of interest
Step2. State null and alternative hypotheses
Step3. Identify a level of significance
Step4. Identify the test statistic
Step5. Compute the value of test statistic and p-value
Step6. State the statistical decision and business conclusion
Winter2013 page#1 QMS202-Business Statistics II Chapter11
Note:
1. Null Hypothesis H o: A statement about the value of a population parameter
Alternative Hypothesis H : -The opposite of the null hypothesis
A
- The hypothesis that the researcher wishes to
support.
[The symbol for Alternative Hypothesis in the textbook is ]
1
The goal of the process is to determine whether there is enough evidence to
infer that the alternative hypothesis is true
2. Level of significance (α): - When a null hypothesis is rejected, the test is
said to be statistically significant at that
significance level
- To determine the rejection region.
- Set by researcher before looking at the data
3. Test Statistics: A value, computed from sample information, used to
determine whether or not to reject the null hypothesis
4. p-value : the probability of observing a test statistic at least as extreme as, or
more extreme than, the one computed given that the null hypothesis
is true
[ i. If the p-value is less th, then reject ;
o
ii. If the p-value is more tha, then do not reject o]
5. There are TWO approaches for drawing a statistical decision
i. Critical value approach
ii. p-value approach
6. There are two possible business conclusions:
i. Conclude that there is enough evidence to support the alternative
hypothesis.
ii. Conclude that there is not enough evidence to support the alternative
hypothesis.
7. Type I error: Rejecting the null hypothesis when it is true
8. Type II error: Do not reject the null hypothesis when it is false
Winter2013 page#2 QMS202-Business Statistics II Chapter11
Rejection region
a. For one-tailed test, use α for calculating critical value
α
right-tailed test left-tailed test
critical value critical value
α
a. For two-tailed test, use for calculating critical values
2
critical value critical value
Risks in Decision Making
Winter2013 page#3 QMS202-Business Statistics II Chapter11
When we use sample data to make decisions about population parameters, there
is always a risk of reaching incorrect conclusions. Two types of errors can
happen in hypothesis testing.
Type I error
1.Occurs when we reject H 0, when it is true.
2.The probability of occurring a type I error is α . This is called the level of
significance of the statistical test.
3.We control the type I error by deciding the value ofα that we want to have of
rejectingH 0, when it is true.
4.Typically,α takes values such as 0.01, 0.05 or 0.1. α is the probability of
rejectingH 0when it is true.
Type II error
1.Occurs when we do not reject H 0when it is false.
2.The probability of a type II error .s
3.One way to reduce type II error is to increase the sample size.
4.Large samples generally permit to detect small differences between
hypothesized values and actual parameters.
The β risk
The probability of committing a type II error i. If the difference between the
hypothesized and the actual value of the population parameter is large, thenβ is
small.
Actual Situation
Statistical Decision
H 0True H 0False
Do not reject H0 Correct decision Type II error
Confidence = 1−α P(Type II error) =
Reject H0 Type I error Correct decision
P(Type I error) =α Power = (1−β )
Winter2013 page#4 QMS202-Business Statistics II Chapter11
Example1
The BigMacBurger restaurant chain claims that the waiting time of all customers
for service is normally distributed, with a population mean of waiting time of 4
minutes and a population standard deviation of 1.4 minutes. The quality
assurance department found in a sample of 60 customers at Yonge and Dundas
that the mean waiting time was 4.1 minutes. Can we conclude that the mean
waiting time is more than 4 minutes? Test at 5% of level of significance.
Calculator Output
1-Sample z Test
μ > 4
z = 0.55328
p = 0.29003
x = 4.1
n = 60
Step1
μ
Let be the population mean of waiting time
Step2
H o μ ≤ 4
H A μ > 4
Step3
Level of significance = 0.05
Step 4
one-sample mean z test
Step5
zstat 0.55328 p-value = 0.29003
x − μ 4.1− 4
zstat σ = 1.4 = 0.55328 p-value = P (z > 0.55328)= 0.29003
n 60
zcritical.64485
Step6
Since the p-value > 0.05, do not reject the null hypothesis.
There is not enough evidence to conclude that the population mean of waiting
time is more than 4 minutes.
Winter2013 page#5 QMS202-Business Statistics II Chapter11
Example2
A recent survey found that young professional working on Bay Street watched an
average of 6.8 DVDs per month. A random sample of 36 young professionals
revelaed that the mean number of DVDs watched last month was 6.2, with a
population standard deviation of 2.5. Can we conclude that young professionals
α
on Bay Street watch fewer than 6.8 DVDs? Use =0.05
Calculator Output
1-Sample z Test
μ
< 6.8
z = -1.44
p = 0.074933
x = 6.2
n = 36
Step1
Let μ
Step2
H o μ
H A μ
Step3
Level of significance =
Step 4
Step5
zstat p-value =
x − μ
zstat σ =
n
zcritical
Step6
Winter2013 page#6 QMS202-Business Statistics II Chapter11
T Test of Hypothesis for the Mean (population standard
deviation σ unknown)
Example3
The Ryerson’s Discount Appliance Store issues its own credit card. The credit
card manager wants to find whether the mean monthly-unpaid balance is more
than $500. The level of significance is set at 0.05. A random check of 180 unpaid
balances revealed the sample mean is $508 and the standard deviation of the
sample is $47. It is known that the monthly-unpaid balance is normally
distributed. Should the credit card manager conclude that the population mean is
greater than $500?
Calculator Output
1-Sample t Test
μ
> 500
t = 2.2836
p = 0.011783
x = 508
sx = 47
n
= 180
Step1
Let μ
Step2
H o μ
H : μ
A
Step3
Level of significance =
Step 4
Step5
t = p-value =
stat
x − μ
t s
stat =
n
d.f = t =
critical
Step6
Winter2013 page#7 QMS202-Business Statistics II Chapter11
Example4
At the time John was hired as a server at the Ryerson Family restaurant at
Dundas and Bay, he was told” You can earn average of not more than $120 a
day in tips.” Over the first 40 days he was employed at the restaurant, the mean
daily amount of his tips was $120.50 and a standard deviation of $41.32. It is
known that the daily amount of tips is normally distributed. At the 0.05
significance level, can John conclude that he is earning an average of more than
$120 in tips per day in the entire year?
Calculator Output
1-Sample t Test
μ > 120
t = 0.076531
p = 0.46969
x
= 120.5
sx = 41.32
n = 40
Step1
Let μ
Step2
H o μ
H A μ
Step3
Level of significance =
Step 4
Step5
tstat p-value =
d.f = t
critical
Step6
Winter2013 page#8 QMS202-Busines

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