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BA 275 Quantitative Business Methods

Confidence Interval Estimation Estimating the Population Proportion

Hypothesis Testing Elements of a Test Concept behind a Test Examples

Agenda

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Quiz #4 : Part I

An automatic machine in a manufacturing process is operating properly if the lengths of an important subcomponent are normally distributed with mean of 117 cm and standard deviation of 5.2 cm.

Find the probability that one selected subcomponent is longer than 120 cm.

Find the probability that if four subcomponents are randomly selected, their mean length exceeds 120 cm.

You, as an inspector, randomly selected 25 subcomponents and measured their mean length to be 120 cm. Based on your sample, do you think the machine is still in control (i.e., the machine is still producing subcomponents with a mean length of 117 cm), or is already out of control (the mean length is no longer 117 cm as required)? Support your argument with numerical evidence.

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Quiz #4: Part II

The mean amount for all of the invoices for your company last month is not known. Based on your past experience, you are willing to assume that the standard deviation of invoice amounts is about $200.

If you take a random sample of 100 invoices, what is the value of the standard deviation for the sample mean ?

The empirical rule says that the probability is about 0.95 that the sample mean is within $A of the population mean m. What is the value of A?

Construct a 95% confidence interval for the true population mean.

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Central Limit Theorem

In the case of sample mean

),(~2

nNX

In the case of sample proportion

))1(

,(~ˆn

pppNp

nX

n

pp

n

ppp

)ˆ1(ˆ)1(ˆ

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100(1-)% Confidence Interval for the Proportion (p.13, formula 3)

n

ppzp

)ˆ1(ˆ2/

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Example (p.14)

A sample of 35 student information sheets shows that 9 intend to concentrate in Finance. Give a 99% confidence interval for the proportion of students in the population that intend to concentrate in Finance.

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Margin of Error (B)

n

ppzp

n

stX

nzX

)ˆ1(ˆˆ 2/

2/

2/

Margin of Error (how good is your point estimate?)

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Estimation in Practice

Determine a confidence level (say, 95%). How good do you want the estimate to be? (define

margin of error) Use formulas (p.14) to find out a sample size that

satisfies pre-determined confidence level and margin of error.

problem. "proportion population" afor )1(

problem. mean" population" afor

2

2/

2

2/

B

ppzn

B

zn

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Example (p.14)

A marketing manager for a start-up firm in Michigan wishes to discover the proportion of teenagers in Japan who own a CD player. If the manager wants a confidence interval of width 0.1, how many teenagers must be sampled? Use a conservative estimate of p. Assume that the confidence level is to be 95%.

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Accuracy Gained by Increasing the Sample Size

Margin of Error (B) Sample Size (n)

7% 196

6% 266

5% 384

4% 600

3% 1037

2% 2401

1% 9604

2

)5.0)(5.0(96.1

Bn

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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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Statistical Inference: Hypothesis Testing

Research Question: Is a claim about the parameter value supported?

Sample of size n

Population

Tool (i.e., formula):Z or T score

Example: = 10,000n = 100Is “ > 22,000”?

Example: “ 22,000”?

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Elements of a Test

Hypotheses Null Hypothesis H0 Alternative Hypothesis Ha

Test Statistic Decision Rule (Rejection Region)

Before collecting dataAfter collecting data

Evidence (actual observed test statistic) Conclusion

Reject H0 if the evidence falls in the R.R. Do not reject H0 if the evidence falls outside the R.R.

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Example

A bank has set up a customer service goal that the mean waiting time for its customers will be less than 2 minutes. The bank randomly samples 30 customers and finds that the sample mean is 100 seconds. Assuming that the sample is from a normal distribution and the standard deviation is 28 seconds, can the bank safely conclude that the population mean waiting time is less than 2 minutes?