Transcript Slides 1-22 Estimation

BA 275
Quantitative Business Methods
Agenda
 Quiz #2
 Sampling Distribution of a Statistic
 Statistical Inference: Confidence Interval Estimation
 Introduction
 Estimating the population mean m
 Examples
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Central Limit Theorem (CLT)
 The CLT applied to Means
If X ~ N ( m ,  2 ) , then X ~ N ( m ,
2
).
n
If X ~ any distribution with a mean m, and variance 2,
then X ~ N ( m ,
2
n
) given that n is large.
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Example 1
 The dean of a B-school claims that the average weekly income
of graduates of her school one year after graduation is $600.
 If the dist. of weekly incomes is normal with a std of $100,


Q1. What is the prob. of one randomly selected graduate
has an avg weekly income of less than $550?
Q2. What is the prob. of 64 randomly selected graduates
have an avg weekly income of less than $550?
 Q3. If the dist. of weekly incomes is UNKNOWN, but the std is
believed to be $100, can we still answer the above two
questions?
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Example 2
An automatic machine in a manufacturing process is
operating properly if the lengths of an important
subcomponent are normally distributed, with mean
117 cm and standard deviation 5.2 cm.
1. Find the probability that one randomly selected
unit has a length greater than 120 cm.
2. Find the probability that, if four units are
randomly selected, their mean length exceeds
120 cm.
3. Find the probability that, if forty units are
randomly selected, their mean length exceeds
120 cm.
4. Without the assumption of normality, can we still
answer Questions 1 – 3.
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Example 3
 The number of cars sold annually by used car
salespeople is normally distributed with a
standard deviation of 15. A random sample
of 400 salespeople was taken and the mean
number of cars sold annually was found to be
75. Find the 95% confidence interval
estimate of the population mean. Interpret
the interval estimate.
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Statistical Inference: Estimation
Population
Example:
 = 10,000
n = 100
What is the value of m?
Research Question:
What is the parameter value?
Example: m =?
Sample of size n
Tools (i.e., formulas):
Point Estimator
Interval Estimator
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100(1-a)% Confidence Interval for the
Mean
 If  is known (or n is large): Section 6.1
X  za / 2

n
 If  is unknown (or n is small): Section 7.1
X  ta / 2
s
n
with degrees of freedom = n - 1
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Practice Problems (Za/2)
 What are the values of za/2 for 86%, 92%, and
97% confidence intervals?
 Which of the three intervals is wider?
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Example 3 (continued)
 The number of cars sold annually by used car
salespeople is normally distributed with a
standard deviation of 15. A random sample
of 400 salespeople was taken and the mean
number of cars sold annually was found to be
75. Find the 90% confidence interval
estimate of the population mean. Interpret
the interval estimate.
 Za/2=?
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Example 4
 Suppose that the amount of time teenagers
spend weekly at part-time jobs is normally
distributed with a standard deviation of 20
minutes. A random sample of 100
observations is drawn and the sample mean
is computed as 125 minutes. Determine the
92% confidence interval estimate of the
population mean.
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Standard Normal Probabilities (Table
A)
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Standard Normal Probabilities (Table
A)
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Answer Key to the Examples used
 Example 1
 Q1: 0.3086; Q2: 0.0000; Q3: only Q2 because of the
central limit theorem
 Example 2
 Q1: 0.281; Q2: 0.1251; Q3: 0.0000; Q4: Yes to all
because the population distribution is normal.
 Example 3: 75 ± 1.96 x (15 / SQRT(400) )
 Practice Problems: For 86%, za/2 = 1.47; For 92%,
za/2 = 1.75; For 97%, za/2 = 2.17
 Example 3: 75 ± 1.645 x (15 / SQRT(400) )
 Example 4: 125 ± 1.75 x (20 / SQRT(100) )
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