Estimating Population Values - Statistical Method - Lecture Slides, Slides of Statistics

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Business Statistics

Estimating Population Values

Business Statistics

Topic Index Probability Distribution Sampling Distribution Point & Interval Estimates

Point and Interval Estimates

• A point estimate is a single number,
• a confidence interval provides additional

information about variability

4

Point Estimate

Lower Confidence Limit

Upper Confidence Limit Width of confidence interval

Point Estimates

5

Estimate Population

Parameters …

with Sample

Statistics

Mean

Proportion

Variance

Difference

μ

p

2

X

P S

2

S

X 1 − X 2

Confidence Interval Estimate

• An interval gives a range of values:

• Takes into consideration variation in

sample statistics from sample to sample

• Based on observation from 1 sample
• Gives information about closeness to

unknown population parameters

• Stated in terms of level of confidence
  • Never 100% sure

Estimation Process

8

(mean, μ, is unknown)

Population

Random Sample

Mean x = 50

Sample

I am 95% confident that μ is between 40 & 60.

Confidence Level

• Confidence Level

– Confidence in which the interval

will contain the unknown

population parameter

• A percentage (less than 100%)

Confidence Level, (1-α)

• Suppose confidence level = 95%
• Also written (1 - α) =.
• A relative frequency interpretation:
  • In the long run, 95% of all the confidence intervals that can be constructed will contain the unknown true parameter
• A specific interval either will contain or

will not contain the true parameter

• No probability involved in a specific interval

Confidence Interval for μ (σ

Known)

• Assumptions
  • Population standard deviation σ is known
  • Population is normally distributed
  • If population is not normal, use large sample
• Confidence interval estimate for μ

13

n

x ± zα/

Finding the Critical Value

• Consider a 95% confidence interval:

14

z. 025 = -1.96 z. 025 = 1.

1 −α =.

. 2

α (^) =. 2

α (^) =

Point Estimate

Lower Confidence Limit

Upper Confidence Limit

z units: x units: (^) Point Estimate

0

zα/2 = ± 1.

Interval and Level of Confidence

16

μx = μ

Confidence Intervals

Intervals extend from

to

100(1-α)% of intervals constructed contain μ; 100 α% do not.

Sampling Distribution of the Mean

n

σ x + zα/

n

σ x − zα/

x

x x

α /2^1 −α α/

Margin of Error

• Margin of Error (e): the amount added and

subtracted to the point estimate to form the

confidence interval

17

n

x ± zα/

n

e = zα/

Example: Margin of error for estimating μ, σ known:

Case Study 4.A

• A sample of 11 circuits from a large normal

population has a mean resistance of 2.20 ohms. We know from past testing that the population standard deviation is .35 ohms.

• Determine a 95% confidence interval for the

true mean resistance of the population.

Solution – Case Study 4.A

• A sample of 11 circuits from a large normal

population has a mean resistance of 2.20 ohms. We know from past testing that the population standard deviation is .35 ohms.

• Solution:

20

1.9932 ............... 2.

2.20.

2.20 1.96(.35/ 11 )

n

σ x z /

= ±

= ±

± (^) α