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BA 275Quantitative Business Methods

Quiz #2

Sampling Distribution of a Statistic

Statistical Inference: Confidence Interval Estimation Introduction Estimating the population mean Examples

Agenda

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Central Limit Theorem (CLT)

The CLT applied to Means

If ),(~ 2NX , then ),(~2

nNX

.

If ~X any distribution with a mean , and variance 2,

then ),(~2

nNX

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

Research Question: What is the parameter value?

Sample of size n

Population

Tools (i.e., formulas):Point EstimatorInterval Estimator

Example: = 10,000n = 100What is the value of ?

Example: ?

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100(1-)% Confidence Interval for the Mean If is known (or n is large): Section 6.1

If is unknown (or n is small): Section 7.1

nzX

2/

n

stX 2/ with degrees of freedom = n - 1

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Practice Problems (Z/2)

What are the values of z/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.

Z/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%, z/2 = 1.47; For 92%,

z/2 = 1.75; For 97%, z/2 = 2.17 Example 3: 75 ± 1.645 x (15 / SQRT(400) ) Example 4: 125 ± 1.75 x (20 / SQRT(100) )