Statistics for Business and Economics, 6/e

Download Report

Transcript Statistics for Business and Economics, 6/e

BUS173 Instructor: Nahid Farnaz (Nhn) North South University

Statistics for Business and Economics

6 th Edition

Chapter 8 Estimation: Single Population

Chap 8-1

Point and Interval Estimates

  A point estimate is a single number, a confidence interval provides additional information about variability

Lower Confidence Limit Point Estimate Width of confidence interval

Upper Confidence Limit

Chap 8-2

Point Estimates

Chap 8-3

Confidence Intervals



Confidence Interval Estimator

for a population parameter is a rule for determining (based on sample information)

a range or an interval

that is likely to include the parameter.

 The corresponding estimate is called a confidence interval estimate.

Chap 8-4

Confidence Interval and Confidence Level

 If P(a <  < b) = 1  then the interval from a to b is called a 100(1 interval of  .  )% confidence  The quantity (1 level  ) is called the confidence of the interval (  between 0 and 1)   In repeated samples of the population, the true value of the parameter  would be contained in 100(1  )% of intervals calculated this way. The confidence interval calculated in this manner is written as a <  < b with 100(1  )% confidence Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

Chap 8-5

Confidence Level, (1-



)

(continued)

   Suppose confidence level = 95% Also written (1  ) = 0.95

A relative frequency interpretation:  From repeated samples, 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 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

Chap 8-6

General Formula

 The general formula for all confidence intervals is:

Point Estimate

(Reliability Factor)(Standard Error)

Chap 8-7

Confidence Intervals

Confidence Intervals Population Mean Population Proportion σ 2 Known σ 2 Unknown

Chap 8-8

Confidence Interval for μ ( σ

Known)

 Assumptions    Population variance σ 2 is known Population is normally distributed If population is not normal, use large sample  Confidence interval estimate: x  z α/2 σ n  μ  x  z α/2 σ n (where z  /2 each tail) is the normal distribution value for a probability of  /2 in Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

Chap 8-9

Margin of Error

 The confidence interval, σ x  z α/2  μ n  x  z α/2 σ n  Can also be written as x  ME where ME is called the margin of error ME  z α/2 σ n  The interval width , w, is equal to twice the margin of error.  w = 2(ME) Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

Chap 8-10

Reducing the Margin of Error

ME  z α/2 σ n The margin of error can be reduced if  the population standard deviation can be reduced ( σ↓)  The sample size is increased (n ↑)  The confidence level is decreased, (1 –  ) ↓ Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

Chap 8-11

Finding the Reliability Factor, z

 /2  Consider a 95% confidence interval: 1    .95

α 2  .025

Z units: X units: z = -1.96

Lower Confidence Limit 0 z = 1.96

Upper Confidence Limit

 Find z .025

Chap 8-12

Common Levels of Confidence

 Commonly used confidence levels are 90%, 95%, and 99%

Confidence Level

80% 90% 95% 98% 99% 99.8% 99.9%

Confidence Coefficient,

1  

.80

.90

.95

.98

.99

.998

.999



/2 value

1.28

1.645

1.96

2.33

2.58

3.08

3.27

Chap 8-13

Example 8.4: Refined Sugar (Confidence Interval)

 A process produces bags of refined sugar. The weights of the content of these bags are normally distributed with standard deviation 1.2 ounces. The contents of a random sample of 25 bags has a mean weight of 19.8 ounces.  Find the upper and lower confidence limits of a 99% confidence interval for the true mean weight for all bags of sugar produced by the process.

Chap 8-14

Example (practice)

 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 0.35 ohms.  Determine a 95% confidence interval for the true mean resistance of the population.

Chap 8-15

Example

(continued)

 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: x  z σ n  2.20

 1.96

(.35/ 11 )  2.20

 .2068

1.9932

 μ  2.4068

Chap 8-16

Interpretation

 We are 95% confident that the true mean resistance is between 1.9932 and 2.4068 ohms  Although the true mean may or may not be in this interval, 95% of intervals formed in this manner will contain the true mean Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

Chap 8-17

Confidence Intervals

Confidence Intervals Population Mean Population Proportion σ 2 Known σ 2 Unknown

Chap 8-18

Student

’

s t Distribution

 Consider a random sample of n observations  with mean x and standard deviation s  from a normally distributed population with mean μ  Then the variable t  x s/  μ n follows the Student ’ s t distribution with (n - 1) degrees of freedom Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

Chap 8-19

Confidence Interval for μ ( σ

Unknown)

 If the population standard deviation σ is unknown, we can substitute the sample standard deviation, s  This introduces extra uncertainty, since s is variable from sample to sample  So we use the t distribution instead of the normal distribution Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

Chap 8-20

  

Confidence Interval for μ ( σ Unknown)

(continued)

Assumptions    Population standard deviation is unknown Population is normally distributed If population is not normal, use large sample Use Student ’ s t Distribution Confidence Interval Estimate: x  t n 1, α/2 S n  μ  x  t n 1, α/2 S n where t n-1, α/2 is the critical value of the t distribution with n-1 d.f. and an area of α/2 in each tail: P(t n  1  t n  1, α/2 )  α/2 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

Chap 8-21

Student

’

s t Distribution

 The t is a family of distributions  The t value depends on degrees of freedom (d.f.)  Number of observations that are free to vary after sample mean has been calculated d.f. = n - 1 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

Chap 8-22

Example 8.5

 Gasoline prices rose drastically during the early years of this century. Suppose that a recent study was conducted using truck drivers with equivalent years of experience to test run 24 trucks of a particular model over the same highway. The sample mean and standard deviation is 18.68 and 1.69526 respectively.

 Estimate the population mean fuel consumption for this truck model with 90% confidence interval.

Chap 8-23

Example (practice)

A random sample of n = 25 has x = 50 and s = 8. Form a 95% confidence interval for μ  d.f. = n – 1 = 24, so t n  1, α/2  t 24,.025

 2.0639

The confidence interval is x  t n 1, α/2 S n 50  (2.0639) 8 25 46.698

 μ    μ μ   x  t n 1, α/2 50  53.302

Chap 8-24

Confidence Intervals

Confidence Intervals Population Mean Population Proportion σ Known σ Unknown

Chap 8-25

Confidence Intervals for the Population Proportion, p

Chap 8-26

Confidence Intervals for the Population Proportion, p

(continued)

 Recall that the distribution of the sample proportion is approximately normal if the sample size is large, with standard deviation σ P  P(1  P) n  We will estimate this with sample data: Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

ˆ (1  ˆ ) n Chap 8-27

Confidence Interval Endpoints

 Upper and lower confidence limits for the population proportion are calculated with the formula  z α/2 ˆ (1  ˆ ) n  P   z α/2 ˆ (1  ˆ ) n  where    z  /2 p is the standard normal value for the level of confidence desired is the sample proportion n is the sample size Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

Chap 8-28

Example

 A random sample of 100 people shows that 25 are left-handed.  Form a 95% confidence interval for the true proportion of left-handers Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

Chap 8-29



Example

(continued)

A random sample of 100 people shows that 25 are left-handed. Form a 95% confidence interval for the true proportion of left-handers.

 z α/2 (1  n )  P   z α/2 ˆ (1  n ) 25 100  1.96

.25(.75)  P 100 0.1651

 P  25 100  1.96

 0.3349

Chap 8-30

Interpretation

 We are 95% confident that the true percentage of left-handers in the population is between 16.51% and 33.49%.  Although the interval from 0.1651 to 0.3349 may or may not contain the true proportion, 95% of intervals formed from samples of size 100 in this manner will contain the true proportion.

Chap 8-31

Statistics for Business and Economics, 6/e

Transcript Statistics for Business and Economics, 6/e

Statistics for Business and Economics

Chapter 8

Estimation: Single Population

Point and Interval Estimates

Point Estimates

Confidence Intervals

Confidence Interval and Confidence Level

Confidence Level, (1-

)

General Formula

Confidence Intervals

Confidence Interval for μ ( σ

Known)

Margin of Error

Reducing the Margin of Error

Finding the Reliability Factor, z

Common Levels of Confidence

Example 8.4: Refined Sugar (Confidence Interval)

Example (practice)

Example

Interpretation

Confidence Intervals

Student

s t Distribution

Confidence Interval for μ ( σ

Unknown)

Confidence Interval for μ ( σ Unknown)

Student

s t Distribution

Example 8.5

Example (practice)

Confidence Intervals

Confidence Intervals for the Population Proportion, p

Confidence Intervals for the Population Proportion, p

Confidence Interval Endpoints

Example

Example

Interpretation

Directory