Transcript Exercise

An importer of Herbs and Spices claims that average weight of packets of Saffron is 20
grams. However packets are actually filled to an average weight, 19.5 grams and
standard deviation, σ=1.8 gram. A random sample of 36 packets is selected, calculate
A- The probability that the average weight is 20 grams or more;
B- The two limits within which 95% of all packets weight;
C- The two limits within which 95% of all weights fall (n=36);
D- If the size of the random sample was 16 instead of 36 how would this affect the
results in (a), (b) and (c)? (State any assumptions made)
Samples
1 and 3
1 and 5
1 and 7
1 and 9
3 and 5
3 and 7
3 and 9
5 and 7
5 and 9
7 and 9
The means of these samples
2
3
4
5
4
5
6
6
7
8
Example
Random samples of size n =2 are drawn from a finite
population that consists of the numbers 2, 4, 6 and 8
without replacement.
a-) Calculate the mean and the standard deviation of this
population
b-) List six possible random samples of size n=2 that can be
drawn from this population and calculate their means.
c-) Use the results in b-) to construct the sampling
distribution of the mean.
d-) Calculate the standard deviation of the sampling
distribution.
Example 2: Tuition Cost
The mean tuition cost at state universities throughout the USA is 4,260 USD per
year (2002 year figures). Use this value as the population mean and assume that
the population standard deviation is 900 USD. Suppose that a random sample of
50 state universities will be selected.
A-) Show the sampling distribution of x̄ (where x̄ is the sample mean tuition cost
for the 50 state universities)
B-) What is the probability that the random sample will provide a sample mean
within 250 USD of the population mean?
C-) What is the probability that the simple random sample will provide a sample
mean within 100 USD of the population mean?
Example 1:
A random variable of size 15 is taken from normal distribution with mean 60 and
standard deviation 4. Find the probability that the mean of the sample is less than 58.
Example 4:
Suppose we have selected a random sample of n=36 observations from a population with
mean equal to 80 and standard deviation equal to 6.
Q: Find the probability that x̄ will be larger than 82.
Example 5: Ping-Pong Balls
The diameter of a brand of Ping-Pong balls is approximately normally distributed, with a
mean of 1.30 inches and a standard deviation of 0.04 inch. If you select a random sample
of 16 Ping-Pong balls,
A-) What is the sampling distribution of the sample mean?
B-) What is the probability that sample mean is less than 1.28 inches?
C-) What is the probability that sample mean is between 1.31 and 1.33 inches?
D-) The probability is 60% that sample mean will be between what two values,
symmetrically distributed around the population mean?
Example 6: E-Mails
Time spent using e-mail per session is normally distributed, with a mean of 8 minutes
and a standard deviation of 2 minutes. If you select a random sample of 25 sessions,
A-) What is the probability that sample mean is between 7.8 and 8.2 minutes?
B-) What is the probability that sample mean is between 7.5 and 8.0 minutes?
C-) If you select a random sample of 100 sessions, what is the probability that sample
mean is between 7.8 and 8.2 minutes?
D-) Explain the difference in the results of (A) and (C).
Types of Survey Errors
• Coverage error
Excluded from frame
• Non response error
• Sampling error
• Measurement error
Follow up on
nonresponses
Random differences from
sample to sample
Bad or leading question
Population
Distribution
?
??
?
Sampling
Distribution
??
? ??
?
Sample
?
?
X
Standard Normal
Distribution
Standardize
x
Z
Sampling Distribution Properties
Larger sample
size
As n increases,
σx
decreases
Smaller sample
size
μ
x
Sampling Distribution
Properties
μx  μ
(i.e.
Normal Population
Distribution
x is unbiased )
Variation:
σ
σx 
n
μ
x
μx
x
Normal Sampling
Distribution
(has the same mean)
How Large is Large Enough?
• For most distributions, n ≥ 30 will give a
sampling distribution that is nearly normal
• For fairly symmetric distributions, n ≥ 15
• For normal population distributions, the
sampling distribution of the mean is always
normally distributed
Example : Paint Supply
The manager of a paint supply store wants to estimate the actual amount of paint
contained in 1-gallon cans purchased from a nationally known manufacturer. The
manufacturer’s specifications state that the standard deviation of the amount of paint
is equal to 0.02 gallon. A random sample of 50 cans is selected, and the sample mean
amount of paint per 1-gallon can is 0.995 gallon.
A-) Construct a 99% Confıdence Interval estimate for the population mean amount of
paint included in a 1-gallon can
B-) On the basis of these results, do you think that the manager has a right to complain
to the manufacturer? Why?
C-) Must you assume that the population amount of paint per can is normally
distributed here? Explain.
D-) Construct a 95% confidence Interval estimate. How does this change your answer
to (B)?
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?
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 – 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
You recently received a job with a company that manufactures an automobile
antitheft device. To conduct an advertising campaign for the product, you need to
make a claim about the number of automobile thefts per year. Since the population
of various cities in the United States varies, you decide to use rates per 10,000
people. (The rates are based on the number of people living in the cities.) Your boss
said that last year the theft rate per 10,000 people was 44 vehicles. You want to see if
it has changed. The following are rates per 10,000 people for 36 randomly selected
locations in the United States. Assume σ = 30.3
55
42
125
62
134
73
39
69
23
94
73
24
51
55
26
66
41
67
15
53
56
91
20
78
70
25
62
115
17
36
58
56
33
75
20
16
Using this information, answer these questions.
1. What hypothesis would you use?
2. Is the sample considered small or large?
3. What assumption must be met before the hypothesis test can be conducted?
4. Which probability distribution would you use?
5. Would you select a one –or two –tailed test? Why?
6. What critical value(s) would you use?
7. Conduct a hypothesis test.
8. What is your decision?
9. What is your conclusion?
10. Write a brief statement summarizing your conclusion.
11. If you lived in a city whose population was about 50,000, how many automobile
thefts per year would you expect to occur?
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 – 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.
Exercise - 1
One Tailed Test
TEST at the 5% level whether the single sample value of 172 comes
from a normal population with mean µ= 150 and variance σ2=100.
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