Transcript Your favorite professional football team (I shall refer to them as the

Exercise - 1
A package-filling process at a Cement company fills bags of cement
to an average weight of µ but µ changes from time to time. The
standard deviation is σ = 3 pounds. A sample of 25 bags has been
taken and their mean was found to be 150 pounds.
Assume that the weights of the bags are normally distributed.
Find the 90% confidence limits for µ.
Exercise - 3
An economist is interested in studying the incomes of consumers in
a particular region. The population standard deviation is known to
be $1,000. A random sample of 50 individuals resulted in an
average income of $15,000.
What is the upper end point in a 99% confidence interval for the
average income?
Exercise - 4
An economist is interested in studying the incomes of consumers in
a particular region. The population standard deviation is known to
be $1,000. A random sample of 50 individuals resulted in an
average income of $15,000.
What is the width of the 90% confidence interval?
Exercise - 5
The head librarian at the Library of Congress has asked her assistant
for an interval estimate of the mean number of books checked out
each day. The assistant provides the following interval estimate:
from 740 to 920 books per day. If the head librarian knows that the
population standard deviation is 150 books checked out per day,
and she asked her assistant for a 95% confidence interval,
approximately how large a sample did her assistant use to
determine the interval estimate?
Exercise – Sports
A researcher hypothesizes that the average number sports that colleges offer for males is greater than
the average number sports that colleges offer for females. A sample of the number of sports offered by
colleges is shown.
At α = 0.10, is there enough evidence to support the claim
Male
Female
8
11
6
11
11
8
15
6
13
8
6
14
8
12
18
7
5
13
14
6
6
9
5
6
9
6
5
5
7
6
6
9
18
7
6
10
7
6
5
5
15
6
11
5
5
16
10
7
8
5
9
9
5
5
8
7
5
5
6
5
8
9
6
11
6
9
18
13
7
10
9
5
11
5
8
7
8
5
7
6
7
7
5
10
7
11
4
6
8
7
10
7
10
8
11
14
12
5
8
5
STEP BY STEP
Critical Value Approach to Hypothesis Testing
1- State Ho and H1
2- Choose level of significance, α
Choose the sample size, n
3- Determine the appropriate test statistics and sampling distribution.
4- Determine the critical values that divide the rejection and non-rejection areas.
5- Collect the sample data, organize the results and compute the value of the test
statistics.
6- Make the statistical decision and state the managerial conclusion
If the test statistics falls into non-rejection region, DO NOT REJECT Ho
If the test statistics falls into rejection region, REJECT Ho
The managerial conclusion is written in the context of the real world problem.
Exercise –Hourly wage
The president of a company states that the average hourly wage of his/her employees is
8.65 TRL. A sample of 50 employees has the distribution shown below. At α=0.05, is the
president’s statement believable? Assume σ=0.105 TRL
Class
8.35-8.43
8.44-8.52
8.53-8.61
8.62-8.70
8.71-8.79
8.80-8.88
Total:
Freq.
2
6
12
18
10
2
50
M
8.39
8.48
8.57
8.66
8.75
8.84
fM
16.78
50.88
102.84
155.88
87.5
17.68
431.56
fM2 _______
140.7842
431.4624
881.3388
1349.9208
765.625
156.2912
3725.4224
Exercise – Athletic Shoes
A researcher claims that the average cost of men`s athletic shoes is less than 80 USD. He
selects a random sample of 36 pairs of shoes from a catalog and finds the following costs. Is
there enough evidence to support the researcher`s claim at α = 0.10. Assume σ=19.2
60
70
75
55
80
55
50
40
80
70
50
95
120
90
75
85
80
60
110
65
80
85
85
45
75
60
90
90
60
95
110
85
45
90
70
70
∑x =2700
Exercise –INFECTIONS
A medical investigation claim that the average
number infections per week at a hospital is 16.3. A
random sample of 10 weeks had a mean number of
17.7 infections. The sample standard deviation is
1.8
Is there evidence to reject the investigator’s claim
at α = 0.05?
Assume the variable is normally distributed
.
Exercise –Internet Access
Z-test for Proportion
Of 2000 adults, 1540 said that they wanted Internet Access so, they could check personal email while on vacation. A survey conducted in the previous year indicated that 75% of adults
wanted Internet Access.
Is there evidence that the percentage of adults who wanted Internet Access has changed from
the previous year
Exercise – Life Guards
A researcher wishes to test the claim that the
average age of lifeguards in a city is greater than 24
years. She selects a sample of 36 guards and finds
the mean of the sample to be 24.7 years with a
standard deviation of 2 years. Is there evidence to
support the claim at α = 0.05?
Use p-value method.
Exercise – Assist. Prof.
A researcher reports that the average salary of
assistant professors is more than 42,000 TL. A
sample of 30 assistant professors has a mean salary
of 43,260 TL. At α = 0.05, Test the claim that
assistant professors earn more than 42,000 TL a
year. The population standard deviation is 5,230 TL.
Exercise – Wind Speed
A researcher claims that the average wind speed in
a certain city 8 miles per hour. A sample of 32 days
has an average wind speed of 8.2 miles per hour.
The standard deviation of the sample is 0.6 mile
per hour. At α = 0.05, is there enough evidence to
reject the claim?
Use p-value method.
Exercise – Starting Salary
A job placement director claims that the average
starting salary for nurses is 24,000 USD. A sample
of 10 nurses` salaries has a mean of 23,450 USD
and a standard deviation of 400 USD. Is there
enough evidence to reject the director`s claim at
α=0.05?
Exercise – Attorney Advertisements
An attorney claims that more than 25% of all
lawyers advertise. A sample of 200 lawyers in a
certain city showed that 63 had used some form of
advertising.
At α = 0.05, is there enough evidence to support
the attorney`s claim?
Use the p-value method.
Exercise – Sugar
Sugar is packed in 5 kg bags. An inspector suspects
the bags may not contain 5 kg. A sample of 50 bags
produces a mean of 4.6 kg and a standard
deviation of 0.7 kg.
Is there enough evidence to conclude that the bags
do not contain 5 kg as stated at α = 0.05?
Also find the 95% CI of the true mean.
A researcher thinks that if expectant mother use vitamin pills, the birth weight of the
babies will increase . The average birth weight of the population is 3.5 kg.
H0:µ = 3.5 and H1 : µ > 3.5
An engineer hypothesizes that the mean number of defects can be decreased in a
manufacturing process of compact disks by using robots instead of humans for certain
tasks. The mean number of defective disks per 1000 is 18.
H0:µ = 18 and H1 : µ < 18
A psychologist feels that playing soft music during a test will change the results of the test.
The psychologist is not sure whether the grades will be higher or lower. In the past, the
mean of the scores was 73.
H0:µ = 73 and H1 : µ ≠ 73
ACCEPT H0
H0 IS TRUE
H0 IS FALSE
REJECT H0
CORRECT DECISION
TYPE I ERROR
(α ERROR)
TYPE II ERROR
(β ERROR)
CORRECT DECISION
If the null hypothesis is true and accepted or false and rejected
the decision is in either case CORRECT.
If the null hypothesis is true and rejected or false and accepted
the decision is in either case in ERROR.
Example : Fast-Food Restaurant
You are manager of a fast-food restaurant. You want to determine whether the waiting
time to place an order has changed in the past month from its previous population mean
value of 4.5 minutes.
A-) State the null Hypothesis and Alternative Hypothesis
From past experience, you can assume that the population is normally distributed with
the standard deviation of 1.2 minutes. You select a sample of 25 orders during one-hour
period. The sample mean is 5.1 minutes.
B- Determine whether there is evidence at the 0.05 level of significance that the
population mean waiting time to place an order has changed in the past month from its
previous population mean value of 4.5 minutes.
C- Find and use p-Value approach to test the Hypothesis.
Exercise –McDonald
One Tailed Test
In one past study, McDonald’s had a mean service time of 174.22 seconds. Suppose that
this company began a quality improvement effort to reduce the service time and selected a
sample of 25 stores. The sample mean has been calculated as 162.96 seconds and sample
standard deviation is 20.2 seconds.
You wish to determine whether the new drive-through process has a mean that is less than
174.22 seconds.
Exercise –Fast Food
One Tailed Test for Proportion
A fast food chain has developed a new process to ensure that orders at the drive-through
are filled correctly. The business problem is defined as determining whether the new
process can increase the percentage of orders processed correctly. The previous process
filled orders correctly 85% of the time.
Data are collected from a sample of 100 orders using the new process. The results indicate
that 94 orders were filled correctly. At the 0.01 level of significance, can you conclude that
the new process has increased the proportion of orders filled correctly?
Exercise - 2
One Tailed Test
TEST at the 1% level whether the single sample value 54 has been
drawn from a normal population with mean 65 and variance 30 or
whether the mean is less than 65.
Exercise – 3
The manager of the women`s dress department of a department
store wants to know whether the true average number of women`s
dresses sold per day is 24.
If in a random sample of 36 days the average number of dresses
sold is 23 with a standard deviation of 7 dresses,
Is there, at the 0.05 level of significance, sufficient evidence to
reject the null hypothesis that µ=24?
Exercise – 4
Exercise – 5
Exercise – 6
Exercise – 7