OPRE504
Chapter Study Guide
Chapter 11
Confidence Intervals and Hypothesis Testing for Means
I.
Calculate Probability for A Sample Mean When Population σ Is Known
1.
First of all, we need to find out the standard deviation of a sampling distribution. When
𝜎
the population’s standard deviation, σ, is known to us, use SD ( 𝑦̅ ) = 𝑛, where n is the sample
√
size.
2.
Calculate z-statistic for a value of sample mean
𝑦̅− 𝜇
z = 𝑆𝐷(𝑦̅) where 𝑦̅ is the sample mean and 𝜇 is the population mean
3.
Find out probability using Z-table (A-32/33 in Appendix C)
Q11.1 The weight of potato chips in a medium-size bag is stated to be 10 ounces (label). The
amount that the packaging machine puts in these bags is believed to have a Normal distribution
N (10.2, 0.12). Answer the following questions [Question 7 for Chapter 11, p.346]:
a) What fraction of all bags sold are underweight?
b) Some of the chips are sold in “bargain packs” of 3 bags. What’s the probability that none of
the 3 is underweight?
c) What’s the probability that the mean weight of the 3 bags is below the stated weight?
d) What’s the probability that the mean weight per bag of a 24-bag case of potato chips is below
10 ounces?
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OPRE504
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Q11.2 Refer to the example in the Textbook (Sharpe 2011, p.324)
II.
Hypothesis Test for Sample Means without Knowing Population’s σ
𝑠
However, since the population parameters are not always known, we can only use SE ( 𝑦̅ ) = 𝑛,
√
to approximate the standard deviation of the sampling distribution, where s is the standard
deviation of the sample we happen to use and n is the sample size and the degree of freedom (df)
is n-1.
1.
State hypotheses (H0 and Ha) and determine whether a two-tailed or one-tailed test
2.
∗
Based on 𝛼 level and degree of freedom (df= n-1), find out the critical value 𝑡𝑛−1
and
determine the rejection region(s) using Student’s T-Table (A-34 in Appendix C)
3.
Calculate standard error of the sampling distribution
𝑠
SE ( 𝑦̅ ) = 𝑛, s = standard deviation of the sample, n = sample size
√
4.
Calculate Student’s t-statistic for the sample
𝑦̅− 𝜇
t = 𝑆𝐸(𝑦̅)
5.
∗
Compare |t| and |𝑡𝑛−1
|:
∗
Reject H0 if |t| ≥ | 𝑡𝑛−1 |;
∗
∗
∗
Fail to Reject H0 if |t| ≤ | 𝑡𝑛−1
| (when t falls in between -𝑡𝑛−1
𝑎𝑛𝑑 𝑡𝑛−1
)
Q11.3 Suppose we would like to know whether the mean GMAT score for all MBA students is
600. Now you take a random sample of 36 MBA students and find that the average GMAT score
for this sample is 640 and the standard deviation for this sample 120. State your hypothesis and
conduct the test. If an alpha level α of 5% is used, what’s your conclusion? If an alpha level α of
10% is used, what is your conclusion?
Step 1: State Hypotheses
Step 2: Find Critical Value Using Two-Tailed T Table
Step 3: Calculate the Standard Error for the Sampling Distribution
Step 4: Calculate Student’s t-statistic for the Sample
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OPRE504
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Step 5: Compare the calculated t-statistic with critical value and make your judgment
(Reject when |t| >| t*|; Fail to reject when |t| ≤ |t*|)
REPEAT STEP 2 AND STEP 5 FOR THE NEW α:
Step 2: Find out the Critical Value associated with α=10% using two-tailed T table
∗
𝑡35,10%
=
Step 5: Compare the calculated t-statistic with critical value and make your judgment
(Reject when |t| > |t*|; Fail to reject when |t| ≤ |t*|)
Q11.4 Suppose we would like to know whether the mean GMAT score for all MBA students is
greater than 600. Now you take a random sample of 36 MBA students and find that the average
GMAT score for this sample is 640 and the standard deviation for this sample 120. State your
hypothesis and conduct the test. If an alpha level α = 5% is used, what’s your conclusion? If an
alpha level α = 1% is used, what is your conclusion?
Step 1: State Hypothesis
Step 2: Find Critical Value Using One-Tailed T-Table
Step 3: Calculate the Standard Error for the Sampling Distribution
Step 4: Calculate Student’s t-statistic for the Sample
Step 5: Compare the calculated t-statistic with critical value and make your judgment
(Reject when |t| > |t*|; Fail to reject when |t| ≤ |t*|)
REPEAT STEP 2 AND STEP 5 FOR THE NEW α:
Step 2: Find out the Critical Value associated with α=1% using one-tailed T-table
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OPRE504
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∗
𝑡35,1%
=
Step 5: Compare the calculated t-statistic with the Critical Value and make your judgment
(Reject when |t| > |t*|; Fail to reject when |t| ≤ |t*|)
Textbook Examples of Hypothesis Tests for Means
Guided Example – Insurance Profits Revisited (pp.334-335)
This is also one-tailed test with lower tail.
III
Confidence Intervals for Means
Model: One-sample t-interval = Estimate ± Marginal Error (ME)
∗
ME = 𝑡𝑛−1
X SE (𝑦̅),
𝑠
∗
CI = 𝑦̅ ± 𝑡𝑛−1
X SE (𝑦̅), SE (𝑦̅) = 𝑛, where n= sample size, df = n-1
√
1.
2.
Determine ∝ level for confidence interval (CI): ∝ = 1-CI
Calculate Standard Error of the Sampling Distribution: SE (𝑦̅) =
3.
4.
Find out the critical value using Two-Tailed T-Table (∝, df= n-1)
𝑠
∗
CI = 𝑦̅ ± 𝑡𝑛−1
X SE (𝑦̅), SE (𝑦̅) = 𝑛
𝑠
√𝑛
√
Q11.5 The average gasoline price per gallon for a random sample of 30 stations in a region is
$4.49 with a standard deviation of $0.29. [Textbook, Q13, p.346].
a)
Find a 95% confidence interval for the mean price of gasoline for all stations in that
region
b)
Find a 90% confidence interval for the mean price of gasoline for all stations in that
region
c)
If we had the sample statistics from a sample of 60 stations, what would be the 95%
confidence interval?
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OPRE504
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IV
Determine Sample Size
CI upper = Mean Estimate + Marginal Error (ME)
CI lower = Mean Estimate - ME
𝑠
ME = CI upper – Mean Estimate SE (𝑦̅) = 𝑛,
√
𝑠
ME = 𝑍 ∗ x 𝑛 (∝= 1- confidence interval) [ Z is used because we don’t know n and need to find
√
it]
𝑍∗
2
n = 𝑀𝐸2 x 𝑠 2
Q11.6 Police departments often try to control traffic speed by placing speed-measuring
machines on roads that tell motorists how fast they are driving. Traffic safety experts must
determine where machines should be placed. In one recent test, police recorded the average
speed clocked by cars driving on one busy street close to an elementary school. For a sample of
25 clocked speeds, it was determined that the average amount over the speed limit for the 25
clocked speeds was 11.6 mph with a standard deviation of 8 mph. The 95% confidence interval
estimate for this sample is 8.30 mph to 14.90 mph [Sharpe 2011, Q57-58, p.353]
a)
What is the margin of error for this problem?
b)
The researchers commented that the interval was too wide. Explain what should be done
to reduce the margin of error to no more than ±2 mph.
c)
To ensure that error rates are estimated accurately, the researchers want to take a large
enough sample that will ensure that usable and accurate interval estimates of how much the
machines may be off in measuring actual speeds. Specifically, the researchers want the margin of
error for a single speed measurement to be no more than ±1.5 mph at the 95% confidence
interval. How may the researchers obtain a reasonable estimate of the standard deviation of error
in the measured speeds?
d)
Suppose the standard deviation of for the error in the measured speeds is believed to be 4
mph from a pilot study, what should be the sample size for next study to ensure that the margin
of error is no larger than ±1 mph.
Chaodong Han
OPRE504
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