sst5e_ppt_ch09.pptx

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STATISTICS

INFORMED DECISIONS USING DATA

Fifth Edition, Global Edition

Chapter 9

Estimating the

Value of a

Parameter

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9.2 Estimating a Population Mean

Learning Objectives

1. Obtain a point estimate for the population mean

2. State properties of Student’s

t

-distribution

3. Determine

t

-values

4. Construct and interpret a confidence interval for a

population mean

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9.2 Estimating a Population Mean

9.2.1 Obtain a Point Estimate for the Population Mean

(1 of 3)

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9.2 Estimating a Population Mean

9.2.1 Obtain a Point Estimate for the Population Mean

(2 of 3)

Parallel Example 1: Computing a Point Estimate

Pennies minted after 1982 are made from 97.5% zinc and 2.5% copper. The following data represent the weights (in grams) of 17 randomly selected pennies minted after 1982.

2.46 2.47 2.49 2.48 2.50 2.44 2.46 2.45 2.49 2.47 2.45 2.46 2.45 2.46 2.47 2.44 2.45

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9.2 Estimating a Population Mean

9.2.1 Obtain a Point Estimate for the Population Mean

(3 of 3)

Solution

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9.2 Estimating a Population Mean

9.2.2 State Properties of Student’s

t

-Distribution

(1 of 9)

Student’s

t

-Distribution

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9.2 Estimating a Population Mean

9.2.2 State Properties of Student’s

t

-Distribution

(2 of 9)

Parallel Example 2: Comparing the Standard Normal

Distribution to the t-Distribution Using Simulation

Obtain 1,000 simple random samples of size n = 5 from a normal population with μ = 50 and σ = 10.

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9.2 Estimating a Population Mean

9.2.2 State Properties of Student’s

t

-Distribution

(3 of 9)

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9.2 Estimating a Population Mean

9.2.2 State Properties of Student’s

t

-Distribution

(4 of 9)

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9.2 Estimating a Population Mean

9.2.2 State Properties of Student’s

t

-Distribution

(5 of 9)

CONCLUSION:

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9.2 Estimating a Population Mean

9.2.2 State Properties of Student’s

t

-Distribution

(6 of 9)

CONCLUSION:

• The histogram for t is also symmetric and bell-shaped with the center of the distribution at 0, but the distribution of t has longer tails (i.e., t is more dispersed), so it is unlikely that t follows a standard normal distribution. The additional spread in the

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9.2 Estimating a Population Mean

9.2.2 State Properties of Student’s

t

-Distribution

(7 of 9)

Properties of the

t

-Distribution

1. The t-distribution is different for different degrees of freedom.

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9.2 Estimating a Population Mean

9.2.2 State Properties of Student’s

t

-Distribution

(8 of 9)

Properties of the

t

-Distribution

5. The area in the tails of the t-distribution is a little greater than the area in the tails of the standard normal distribution,

because we are using s as an estimate of σ, thereby introducing further variability into the t- statistic.

6. As the sample size n increases, the density curve of t gets

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9.2 Estimating a Population Mean

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9.2 Estimating a Population Mean

9.2.3 Determine

t

-Values

(1 of 3)

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9.2 Estimating a Population Mean

9.2.3 Determine

t

-Values

(2 of 3)

Parallel Example 3: Finding

t

-values

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9.2 Estimating a Population Mean

9.2.3 Determine

t

-Values

(3 of 3)

Solution

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9.2 Estimating a Population Mean

9.2.4 Construct and Interpret a Confidence Interval for the

Population Mean

(1 of 11)

Constructing a (1 −

α

)100%

Confidence Interval for

μ

Provided

• sample data come from a simple random sample or randomized experiment,

• sample size is small relative to the population size (n ≤ 0.05N), and

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9.2 Estimating a Population Mean

9.2.4 Construct and Interpret a Confidence Interval for the

Population Mean

(2 of 11)

Constructing a (1 −

α

)100%

Confidence Interval for

μ

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9.2 Estimating a Population Mean

9.2.4 Construct and Interpret a Confidence Interval for the

Population Mean

(3 of 11)

Parallel Example: Using Simulation to Demonstrate the

Idea of a Confidence Interval

We will use Minitab to simulate obtaining 30 simple random

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9.2 Estimating a Population Mean

9.2.4 Construct and Interpret a Confidence Interval for the

Population Mean

(4 of 11)

SAMPLE

MEAN

95.0% CI

C1

47.07 (40.14, 54.00)

C2

49.33 (42.40, 56.26)

C3

50.62 (43.69, 57.54)

C4

47.91 (40.98, 54.84)

C5

44.31 (37.38, 51.24)

C6

51.50 (44.57, 58.43)

C7

52.47 (45.54, 59.40)

C8

59.62 (52.69, 66.54)

C9

43.49 (36.56, 50.42)

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9.2 Estimating a Population Mean

9.2.4 Construct and Interpret a Confidence Interval for the

Population Mean

(5 of 11)

SAMPLE

MEAN

95.0% CI

C11

50.08 (43.15, 57.01)

C12

56.37 (49.44, 63.30)

C13

49.05 (42.12, 55.98)

C14

47.34 (40.41, 54.27)

C15

50.33 (43.40, 57.25)

C16

44.81 (37.88, 51.74)

C17

51.05 (44.12, 57.98)

C18

43.91 (36.98, 50.84)

C19

46.50 (39.57, 53.43)

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9.2 Estimating a Population Mean

9.2.4 Construct and Interpret a Confidence Interval for the

Population Mean

(6 of 11)

SAMPLE

MEAN

95.0% CI

C21

48.75 (41.82, 55.68)

C22

51.27 (44.34, 58.20)

C23

47.80 (40.87, 54.73)

C24

56.60 (49.67, 63.52)

C25

47.70 (40.77, 54.63)

C26

51.58 (44.65, 58.51)

C27

47.37 (40.44, 54.30)

C28

61.42 (54.49, 68.35)

C29

46.89 (39.96, 53.82)

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9.2 Estimating a Population Mean

9.2.4 Construct and Interpret a Confidence Interval for the

Population Mean

(7 of 11)

Note that 28 out of 30, or 93%, of the confidence intervals

contain the population mean

μ

= 50.

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9.2 Estimating a Population Mean

9.2.4 Construct and Interpret a Confidence Interval for the

Population Mean

(8 of 11)

Parallel Example 4: Constructing a Confidence Interval

about a Population Mean

Construct a 99% confidence interval about the population mean weight (in grams) of pennies minted after 1982. Assume μ = 0.02 grams.

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9.2 Estimating a Population Mean

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9.2 Estimating a Population Mean

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9.2 Estimating a Population Mean

9.2.4 Construct and Interpret a Confidence Interval for the

Population Mean

(11 of 11)

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9.2 Estimating a Population Mean

9.2.5 Determine the Sample Size Needed to Estimate the

Population Mean within a Specified Margin of Error

(1 of 3)

Determining the Sample Size

n

The sample size required to estimate the population mean, µ, with a level of confidence (1 − α)·100% with a specified margin of