Transcript document

QMS202
Jason Yim
Jessica Shute
Normal distribution
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
Normal distribution question
Z-test question
Confidence Intervals
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2800 students asked if they’re comfortable to
report cheating by their fellow students. 1344
said Yes, 1456 said No. What is the
confidence Interval for the proportion of
student population who feel comfortable
reporting cheating?
Sample size
Sample size determination for the Mean
 Suppose you want to estimate the population mean to break a
wooden board to within ±20 kilos with 99% confidence . The
standard deviation is 80 kilos. Find the sample size!
 To find Z: 99% confidenceSTAT, DIST, NORM, INVN Area:
0.005, σ=1, µ=0  EXECUTE!
 Implement sample size determination for the mean equation!
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n = 106.15
Sample Size
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Sample size determination for the Proportion
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If there is no information available, plug in the
following numbers to the equation: e=0.05,
π=0.5
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Example: In a survey of 500 people, you find that 60% of Facebook
users will use the “Like” button on facebook. Determine the sample
size needed to estimate the proportion of facebook users who use
the “Like” to within ±0.07 with a 94% confidence
To find Z: 99% confidenceSTAT, DIST, NORM, INVN Area: 0.03,
σ=1, µ=0  EXECUTE!
N= 173.24
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Types of Errors
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Type I error
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If you reject the null hypothesis H0 when it is true
and should not be rejected
Probability of Type I error = 
Type II error
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
If you do not reject the null hypothesis H0 when it
is false and should be rejected
Probability of Type II error = 
Hypothesis Testing and Decision
Making
Actual Situation
Statistical Decision
H0 True
H0 False
Do not reject H0
Correct decision
Confidence = 1 - 
Type II error
P(Type II error) = 
Reject H0
Type I error
P(Type I error) = 
Correct decision
Power = 1 - 
When do we conduct one and
two-tail tests?
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A two-tail test is conducted whenever the alternative hypothesis specifies
that the mean is not equal to the value stated in the null hypothesis
 H0: µ = µ0
 H1: µ ≠ µ0
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There are two types of one-tail tests:
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A one-tail test that focuses on the right tail is conducted whenever the
alternative hypothesis states that the mean is greater than the value
stated in the null hypothesis
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H0: µ = µ0
H1: µ > µ0
A one-tail test that focuses on the left tail is conducted whenever the
alternative hypothesis states that the mean is less than the value stated
in the null hypothesis
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H0: µ = µ0
H1: µ < µ0
1-Sample Z-test
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Assume Normal distribution
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Example: Boozehead Co. produces 40oz beers, and
they’re suspected of cheating their consumers for their
beer. They believe that they’re getting less than 40oz. A
bunch of Boozehead beer drinkers approached head
office who has found out that the bottling process that
fills this type of bottle has a standard deviation of 1.5oz.
For evident reasons, a sample of 48 beers were
measured, and they found that the mean volume was
39.7oz. With a 95% confidence level, does this evidence
support the assumption of Boozehead beer drinkers?
6 Step Method of Hypothesis
Testing
1.
State the null hypothesis H0 and alternative hypothesis H1
2.
Choose the level of significance, , and sample size, n
3.
Determine the appropriate test statistic and sampling distribution
4.
Determine the critical values that divide the rejection and non rejection regions
5.
Collect the data and compute the value of the test statistic
6.
Make the statistical decision and state the managerial conclusion.
If the test statistic falls into the non-rejection region, you do not reject the null
hypothesis.
If the test statistic falls into the rejection region, reject the null hypothesis.
***** Put in sample question
5 Step Method of Hypothesis
Testing Using P-Value
1.
State the null hypothesis, H0, and the alternative hypothesis, H1
2.
Choose the level of significance, , and sample size, n
3.
Determine the appropriate test statistic and sampling distribution
4.
Collect the data, compute the value of the test statistic, and compute the p-value
5.
Make the statistical decision and state the managerial conclusion
If the p-value ≥ , you do not reject the null hypothesis H0
If the p-value < , reject the null hypothesis H0
Recall: if p-value is low, H0 must go
***** Put in sample question from textbook
One-Tail Tests
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A company that manufactures Basketballs is concerned that the
mean weight doesn’t exceed 20 ounces. Standard deviation is 0.04
ounces. A sample of 40 basketballs were selected and the sample
mean is 20.12 ounces. With 99% confidence level, is there evidenec
that the population mean weight of the basketballs is greater than 20
ounces?
Matched Pairs t-test
Can more men get into Night Clubs when
they bring ladies in with them? To
investigate this possibility, a random sample
of 6 clubs are observed at the line-up to see
how many men get into the club with ladies,
and without ladies.
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At 0.01 level of significance, is there
evidence of that the men admitted to clubs
between men coming in alone greater than
men bringing in women to clubs?
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Is p-value >0.01?
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Step 1: Enter a list of the difference between
the numbers.
You can solve for two ways! P-value or T-test
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Step 2: t_calc = t_6-1, 0.01= -3.3649, t_crit =
-1.6682, We reject Ho!
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Step 2: p-value >0.01? Yes! Do not reject
Ho!
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There is enough evidence to conclude that
there aren’t more men coming in alone than
men bringing in women to clubs
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Clubs
Men
Men bringing in
women to the
club
Circa
270
483
Republik 352
389
Lot122
145
150
Guv
156
400
Mckey
Finns
Revival
206
278
216
145
1-Sample t Test
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Unknown standard deviation
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Boozehead Co. produces beer bottles that hold 8oz of beer.
Boozehead…. AGAIN gets complaints that their consumers get less
than 8oz. They sample 22 bottles and finds the average amount of
liquid held by bottles is 7.802 with a std dev of 0.302. Is there
evidence that Boozehead Co. offers their consumers less than 8oz
of beer?
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2-sample t-test
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Comparing the means of two
independent populations
Jack and Jill decided to play
Paper Toss, to see who can
throw in more paper balls into
the garbage can within a
certain time. Jack and Jill did
10 trials and ended up with the
following results. At the 0.05
level of significance, is there
evidence that the Paper Toss
mean is lower for Jack than it
is for Jill?
Ho: µ
Jack
6
8
3
2
6
4
1
8
9
8
Jill
5
7
9
2
3
5
2
6
3
1
1 sample z-test proportion
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A new vending machine was put into the Ted Rogers Business
Building to make sure that it wasn’t eating people’s money. The
previous vending machine was successful to complete
transactions 85% of the time. A sample of 200 vending machine
transactions were observed, and found that 195 orders
completed the transaction successfully. With the 0.02 level of
significance, can you conclude that the new vending machine
has increased the proportion of successful completed
transactions?
Ho: π = 0.85 (the proportion of successfully completed orders is
less or equal to 0.85)
H_1: π >0.85 ( the proportion of orders filled correctly is greater
than 0.85)
2-sample Z-test
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A company makes cigarettes out of two machines. There have
been complains that the some cigarettes were shorter in length
and getting ripped off for what they were paying. In the past, it
has been determined by past studies that the standard deviation
of the cigarette length is 1.5cm from Machine 1, and 1.9 from
Machine 2. The manufacturer decided to take samples of the
cigarettes from both machines to test to see weather the mean
length of the cigarettes from machine 2 was significantly larger
than the mean length in machine 1. The sample of 100 cigarettes
from Machine 1 had a mean length of 6.94cm, and Machine 2
had a mean length of 6.92. At 10% level of significance what is
the conclusion?
F-test for the difference between 2
variances
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Finding Lower-Tail Critical Values
Numerator and denominator for F_U and F_L
are switched around!
F-test for the difference
between 2 variances
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A engineering team has developed a new machine
to assemble car engines, and they want to know if
this new machine has reduced the time to assemble
the car engine. They have taken samples of 60
engines from the new machines, and 50 engines
from the old machine. The mean is 230 minutes,
and the standard deviation is 4 minutes for the new
machines. For the old machine, the mean is 226
minutes with a standard deviation of 6 minutes.
They want to test to see if the assumption about
equal variances seems reasonable at the 5% level.
One-way ANOVA
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At the Italian restaurant,
Westside Marios, a study
was done to see if there is
evidence whether there is a
difference in mean time to
serve food from the
following 4 entrees.
At the 0.05 level of
significance, is there
evidence of a difference in
the mean time to serve food
from the 4 entrees?
Entrees
Minutes
Gnocchi
10
Ravioli
12
Spaghetti
7
Linguine
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Sample Midterm
1
a)
b)
c)
d)
The standard error of the mean
is never larger than the standard deviation of the population
measure the variability of the mean from sample to sample
decreases as sample size increases
all of the above
For air travelers, one of the biggest complaints is from the waiting time between when the airplane taxis away
from the terminal until the time it takes off. This waiting time is known to have a skewed-right distribution with
a mean of 10 minutes and a standard deviation of 8 minutes. Suppose 100 flights have been randomly
sampled. Describe the sampling distribution of the mean waiting time between when the airplane taxis away
from the terminal until the flight takes off for these 100 flights.
2
a)
b)
c)
d)
Distribution is skewed-right with mean = 10 minutes and standard error = 8 minutes
Distribution is approximately normal with mean = 10 minutes and standard error = 0.8 minutes
Distribution is skewed-right with mean = 10 minutes and standard error = 0.8 minutes
Distribution is approximately normal with mean = 10 minutes and standard error = 8 minutes
If the standard error of the sampling distribution of the sample proportion is 0.0229 for samples of size 400,
then the population proportion must be either:
3
a)
b)
c)
d)
0.4 or 0.6
0.5 or 0.5
0.3 or 0.7
0.2 or 0.8
When determining the sample size for a proportion for a given level of confidence and sampling error, the
closer to 0.50 that π is estimated to be, the sample size required
4
a)
b)
c)
d)
is not affected
is larger
is smaller
can be smaller, larger or unaffected
A 99% confidence interval estimate can be interpreted to mean that
5
a)
b)
c)
d)
if all possible samples are taken and confidence interval estimates are developed 99% of them would include the true
population mean somewhere within their interval
we have 99% confidence that we have selected a sample whose interval does include the population mean
both of the above
none of the above
If you were constructing a 99% confidence interval of the population mean based on a sample of n=25,
where the standard deviation of the sample s = 0.05, what will the critical value of t be?
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a)
b)
c)
d)
2.7874
2.4851
2.7969
2.4922
A confidence interval ws used to estimate the proportion of statistics students that are females. A random
sample of 72 statistics students generated the following 90% confidence interval: (0.438, 0.642). Based on
the interval above, is the population proportion of females equal to 0.60?
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a)
b)
c)
d)
Maybe, 0.60 is a believable value of the population proportion based on the information above
No and we are 90% sure of it
No, the population is 54.17%
Yes and we are 90% sure of it
8
As an aid to the establishment of personnel requirements, the director of a hospital wishes to estimate the mean
number of people who are admitted to the emergency room during a 24-hour period. The director randomly
selects 64 different 24-hour periods and determines the number of admissions for each. For this sample, mean
= 19.8 and s2 = 25. Using the sample standard deviation as an estimate for the population standard deviation,
what size sample should the director choose if she wishes to estimate the mean number of admissions per 24hour period to within 1 admission with 98% reliability?
a)
b)
c)
d)
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106
135
136
105
A university dean is interested in determining the proportion of students who receive some sort of financial aid.
Rather than examine the records for all students, the dean randomly selects 200 students and finds that 118 of
them are receiving financial aid. If the dean wanted to estimate the proportion of all students receiving financial
aid to within 3% with 99% reliability, how many students would need to be sampled?
a)
b)
c)
d)
n = 1,843
n=1
n = 1,504
n = 1,784
10
True/False: The t distribution approaches the standardized normal distribution when the number of degrees of
freedom increases.
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True/False: The managers of a company are worried about the morale of their employees. In order to
determine if a problem in this area exists, they decide to evaluate the attitudes of their employees with a
standardized test. They select the Fortunato test of job satisfaction which has a known standard deviation of 24
points.
Referring to the above information, this confidence interval is only valid if the scores on the Fortunato test are
normally distributed.
A hotel chain wants to estimate the average number of rooms rented daily in each month. The population of
rooms rented daily is assumed to be normally distributed for each month with a standard deviation of 24
rooms.
Referring to the above information, during January, a sample of 16 days has a sample mean of 48 rooms.
This information is used t o calculate an interval estimate for the population mean to be from 40 to 56 rooms.
What is the level of confidence of this interval?
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a)
b)
c)
d)
13
95.0%
81.8%
49.5%
26.1%
A university wanted to find out the percentage of students who felt comfortable reporting cheating by their
fellow students. A surveyed of 2,800 students was conducted and the students were asked if they felt
comfortable reporting cheating by their fellow students. The results were 1,344 answered “yes” and 1,456
answered “no”.
Referring to the above information, a 99% confidence interval for the proportion of student population who feel
comfortable reporting cheating by their fellow students of from _____ to _____.
a)
b)
c)
d)
49.6% to 54.4%
45.6% to 50.4%
46.1% to 49.9%
50.1% to 53.9%
A major DVD rental chain is considering opening a new store in an area that currently does not have any such
stores. The chain will open if there is evidence that more than 5,000 of the 20,000 households in the area are
equipped with DVD players. It conducts a telephone poll of 300 randomly selected households in the area
and finds 96 have DVD players. State the test of interest to the rental chain.
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a)
b)
Ho: π = 0.25 versus H1: π > 0.25
H0: π = 0.32 versus H1: π > 0.32
c) H0: π = 5,000 versus H1: π > 5,000
d) H0: µ = 5,000 versus H1: µ > 5,000
A major DVD rental chain is considering opening a new store in an area that currently does not
have any such stores. The chain will open if there is evidence that more than 5,000 of the
20,000 households in the area are equipped with DVD players. It conducts a telephone poll of
300 randomly selected households in the area and finds that 96 have DVD players. The pvalue associated with the test statistic in this problem is approximately equal to
15
a)
b)
c) 0.0051
d) 0.0013
An entrepreneur is considering the purchase of a coin-operated laundry machine. The current
owner claims that over the past 5 years, the average daily revenue was $675 with a standard
deviation of $75. A sample of 30 days reveals a daily average revenue of $625. If you were to
test the null hypothesis that the daily average revenue was $675, which test would you use?
16
a)
b)
t-test of a population proportion
Z-test of a population mean
c) Z-test of a population proportion
d) t-test of a population mean
An entrepreneur is considering the purchase of a coin-operated laundry machine. The current
owner claims that over the past 5 years, the average daily revenue was $675 with a standard
deviation of $75. A sample of 30 days reveals a daily average revenue of $625. If you were to
test the null hypothesis that the daily average revenue was $675 and decide not to reject the
null hypothesis, what would you conclude?
17
a)
b)
c)
d)
18
0.0100
0.0026
There is not enough evidence to conclude that the daily average revenue was not $675
There is enough evidence to conclude that the daily average revenue was $675
There is enough evidence to conclude that the daily average revenue was not $675
There is not enough evidence to conclude that the daily average revenue was $675
True/False: The test statistic measures how close the computed sample statistic has come to
the hypothesized population.
True/False: Suppose we wish to test H0: µ = 8 versus H1 < 8. What will result if we conclude that the mean
is less than 8 when its true value is really 12?
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a)
b)
c)
d)
We have made a Type I error
We have made a Type II error
We have made the correct decision
We do not have enough information
True/False: Suppose we wish to test H0: π = 75% versus H1 > 75%. What will result if we conclude that
the proportion is not greater than 75% when its true value is really 85%?
20
a)
b)
c)
d)
We have made a Type II error
We have made a Type I error
We do not have enough information
We have made the correct decision
Fill in the following table for the next 2 questions:
Null:
Assumptions:
Alternative:
Test Statistic
Comparison
Conclusion
21
A major home improvement store conducted its biggest brand recognition campaign in the company’s
history. A series of new television advertisements featuring well-known entertainers and sports
figures were launched. A key metric for the success of television advertisements is the proportion of
viewers who, “like the ads a lot.” A study of 1,189 adults who viewed the ads reported that 230 indicated
that they, “like the ads a lot.” The percentage of a typical television advertisement receiving the, “like
the ads a lot,” score is believed to be 22%. Company officials wanted to know if there is evidence that the
series of television advertisements are less successful than the typical ad at a 3% level of significance.
Use the above information and the appropriate set of hypothesis testing to determine there is evidence
that the series of television advertisements are less successful than the typical ad at 3% level of
significance. State any necessary assumptions.
THINGS TO REMEMBER

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Know when standard deviation is known or unknown to use the
appropriate testing (Z or t?, sample standard deviation or not?)
Recognize the difference between matched t-test (independent or
dependent)
Easy to evaluate p-value! If p-value is greater than its level of
significance, we DO NOT REJECT H_o, if p-value is less than its
level of significance, we REJECT Ho!
Draw out charts to see where your t, z, and f values are!
Write in assumptions for easy marks
Don’t confuse null hypothesis, and alternative hypothesis
Always follow the 5 steps!
Understsand calculator implementations and what they all mean!
GOOD LUCK!
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SUPPORTING
RYERSON SOS!