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Chapter 7: Confidence Interval and Sample Size Learning Objectives Upon successful completion of Chapter 7, you will be able to: • •

Find the confidence interval for the mean, proportion, and variance. Determine the minimum sample size when determining a confidence interval for the mean and for a proportion.

•

Level of confidence, maximum error of Estimate (E) and the sample size are inter-related.

I. Inference Includes: 1. Estimation of a population parameter (μ, ρ, or ) using data from a sample. 2. Hypothesis Testing or using sample data to test a conjecture about the population mean (μ), population proportion (ρ), or population standard deviation ( ).

II. Two Kinds of Estimate for Parameters 1. A point estimate of the population parameter is the sample statistic, i.e., the point estimate for the population mean μ is the sample mean of , the point estimate for the population proportion is the sample proportion, and the point estimate for the population standard deviation is the sample standard deviation s. 2. An interval estimate of a parameter is a range of values determined from the point estimate.

Dr. Janet Winter, [email protected]

Stat 200

Page 1

III. Confidence Interval Estimates for Population Parameters The confidence level is the probability that intervals determined by these methods will contain the parameter. A confidence interval is the range of values determined from a sample statistic and the specified confidence level. The common confidence intervals use 90%, 95%, or 99% confidence levels.

IV.Confidence Interval Estimates for the Population Mean μ A. When to use the Normal Distribution (z) and when to use the t Distribution for Confidence Interval Estimates of the Population Mean Start

Yes

Yes

Is the population normally distributed?

Yes

z

Use the normal distribution

No

Is σ known? Yes

No

Is n > 30?

Is the population normally distributed?

No

Yes

Use nonparametric or bootstrapping methods.

t Use the t distribution

No

Is n > 30?

No

Use nonparametric or bootstrapping methods.

“Elementary Statistics: Using the Graphing Calculator for the TI-83/84”, Triola, Mario F.

Dr. Janet Winter, [email protected]

Stat 200

Page 2

B. Rounding Rules for all Confidence Intervals Estimates of the Mean I. When using actual data: a) find the mean and standard deviation to 2 extra places than the data. b) round the answer to one more decimal place than the original data. Note: This is very important! Answers not rounded correctly are marked wrong on Mathzone. II. When using a mean and standard deviation, work with one more decimal place than the data and round to the same number of decimal places given for the mean.

C. Meaning of ALL Confidence Interval Estimates Be sure to reread P 353 (6th edition) or P 361 (7th edition) in the textbook to better understand the meaning of the confidence interval. For example: a 90% confidence interval estimate for the population mean is interpreted as 90% of the confidence interval estimates formed with this process include the value of the population mean.

D. z Interval Estimates for the population Mean

I. Requirements a) the population standard deviation ( ) is given b) the sample size n 30; c) But, if the sample size n < 30, the variable must be selected from a normal distribution II. Confidence Coefficient

Dr. Janet Winter, [email protected]

Stat 200

Page 3

a) Meaning of the Confidence Coefficient z is called the confidence coefficient, i.e., the number of multiples of the standard error for an interval estimate with a level of confidence. Complete the rest of the table using the confidence level (1-∝). The first 2 have been completed for you (answers at the end). .

.90 .95

.10 .05

.05 .025

b) Method to find the Confidence Coefficient: Find the z value with area to its left, i.e., 1. Locate inside the Normal Probability Table (Table E) 2. Starting at , move your hand to the left along the row until you reach the Z column. This is the integer and tenths digits. Go back to , next move your hand to the top of its column. This is the hundreds digits. 3.

Add the integer and tenths digits to the hundredths digits to find the value for z.

4.

Affix a

sign in front of the number.

Dr. Janet Winter, [email protected]

Stat 200

Page 4

Using the method described, complete the table below. The first 2 have been completed for you (answer at the end). Confidence Level 1−α

α

.90 .95 .99 .80 .98 .96 .93

.10 .05

α/2

(1 − α ) +

.95 .975

α 2

.95 .975

Confidence Coefficient 𝑧(𝑎/2) 1.645 1.96

III. Development of the Confidence Interval Formula 𝜎 𝜎 𝑥̅ − 𝑧 < 𝜇 < 𝑥̅ + 𝑧 √𝑛 √𝑛

Whenever the population standard deviation 𝜎 is known and either the population is normally distributed or n ≥ 30, the Central Limit Theorem guarantees the sample mean is normally distributed or: 𝑥̅ − 𝜇 −(−𝜇) > −(−𝑥̅ + 𝑧𝜎𝑥̅ ) 𝑥̅ + 𝑧 ∙ 𝜎𝑥̅ > 𝜇 > 𝑥̅ − 𝑧𝜎𝑥̅ 𝑥̅ − 𝑧 ∙ 𝜎𝑥 < 𝜇 < 𝑥̅ + 𝑧𝜎𝑥̅ 𝑥̅ − 𝑧

𝜎

√𝑛

< 𝜇 < 𝑥̅ + 𝑧

𝜎

√𝑛

Note: If the population standard deviation is not known or stated, use x� − t

s

√n

< 𝜇 < x� + t

s

√n

Dr. Janet Winter, [email protected]

(see section E page 9).

Stat 200

Page 5

IV. Review of Concepts and Maximum Error of Estimate is the point estimate and the center of the confidence interval z is the confidence coefficient, the number of multiples of the standard error needed to construct an interval estimate of the correct width to have a level of confidence 1− α

is called the maximum error of estimate. V. Example: 35 fifth-graders have a mean reading score of 82. The standard deviation of the population is 15. a) Find the 95% confidence interval estimate for the mean reading scores of all fifthgraders. Since we know the population standard deviation and n≥30, use . Use Table E backwards with the area to the left of z equal to .025. The value of z or the confidence coefficient is z = 1.9 + .06 = 1.96. This means approximately 95% of the sample means will fall within 1.96 standard errors of the population mean. Use z = 1.96 in the formula.

4.97, rounded to 5, is the maximum error of estimate. Be sure to list it for full credit in your answers.

Dr. Janet Winter, [email protected]

Stat 200

Page 6

b) Find the 99% confidence interval estimate of the mean reading scores of all fifthgraders. Since approximately 99% of the sample means will fall within 2.58 standard errors of the population mean, use z = 2.58 𝑋� = 82.1, 𝑛 = 35, 𝜎 = 15 𝑋� − 𝑧

𝜎

√𝑛

< 𝜇 < 𝑋� + 𝑧

82.1 − 2.58

15

√35

𝜎

√𝑛

< 𝜇 < 82.1 + 2.58

82.1 − 6.54 < 𝜇 < 82.1 + 6.54

15

√35

82.1 ± 6.5

75.6 < 𝜇 < 88.5

6.54, rounded to 6.5, is the maximum error of estimate. Be sure to list it in the next to last step. c) Is the 95% confidence interval or the 99% confidence interval larger? Explain why. 95% confidence level: 77 < μ < 87 99% confidence level: 75 < μ < 89 The 99% confidence level is larger because it has a larger z value.

Question 1 A study of 40 English composition professors showed that they spent, on average, 12.6 minutes correcting a student’s term paper. Find the 90% confidence interval of the mean time for all composition papers when 𝜎 = 2.5 minutes. n = 40 𝑋�= 12.6

Since the population standard deviation is given and n = 40 is greater than 30, use the formula: 𝝈 𝝈 �−𝒛 �+𝒛

View more...
Find the confidence interval for the mean, proportion, and variance. Determine the minimum sample size when determining a confidence interval for the mean and for a proportion.

•

Level of confidence, maximum error of Estimate (E) and the sample size are inter-related.

I. Inference Includes: 1. Estimation of a population parameter (μ, ρ, or ) using data from a sample. 2. Hypothesis Testing or using sample data to test a conjecture about the population mean (μ), population proportion (ρ), or population standard deviation ( ).

II. Two Kinds of Estimate for Parameters 1. A point estimate of the population parameter is the sample statistic, i.e., the point estimate for the population mean μ is the sample mean of , the point estimate for the population proportion is the sample proportion, and the point estimate for the population standard deviation is the sample standard deviation s. 2. An interval estimate of a parameter is a range of values determined from the point estimate.

Dr. Janet Winter, [email protected]

Stat 200

Page 1

III. Confidence Interval Estimates for Population Parameters The confidence level is the probability that intervals determined by these methods will contain the parameter. A confidence interval is the range of values determined from a sample statistic and the specified confidence level. The common confidence intervals use 90%, 95%, or 99% confidence levels.

IV.Confidence Interval Estimates for the Population Mean μ A. When to use the Normal Distribution (z) and when to use the t Distribution for Confidence Interval Estimates of the Population Mean Start

Yes

Yes

Is the population normally distributed?

Yes

z

Use the normal distribution

No

Is σ known? Yes

No

Is n > 30?

Is the population normally distributed?

No

Yes

Use nonparametric or bootstrapping methods.

t Use the t distribution

No

Is n > 30?

No

Use nonparametric or bootstrapping methods.

“Elementary Statistics: Using the Graphing Calculator for the TI-83/84”, Triola, Mario F.

Dr. Janet Winter, [email protected]

Stat 200

Page 2

B. Rounding Rules for all Confidence Intervals Estimates of the Mean I. When using actual data: a) find the mean and standard deviation to 2 extra places than the data. b) round the answer to one more decimal place than the original data. Note: This is very important! Answers not rounded correctly are marked wrong on Mathzone. II. When using a mean and standard deviation, work with one more decimal place than the data and round to the same number of decimal places given for the mean.

C. Meaning of ALL Confidence Interval Estimates Be sure to reread P 353 (6th edition) or P 361 (7th edition) in the textbook to better understand the meaning of the confidence interval. For example: a 90% confidence interval estimate for the population mean is interpreted as 90% of the confidence interval estimates formed with this process include the value of the population mean.

D. z Interval Estimates for the population Mean

I. Requirements a) the population standard deviation ( ) is given b) the sample size n 30; c) But, if the sample size n < 30, the variable must be selected from a normal distribution II. Confidence Coefficient

Dr. Janet Winter, [email protected]

Stat 200

Page 3

a) Meaning of the Confidence Coefficient z is called the confidence coefficient, i.e., the number of multiples of the standard error for an interval estimate with a level of confidence. Complete the rest of the table using the confidence level (1-∝). The first 2 have been completed for you (answers at the end). .

.90 .95

.10 .05

.05 .025

b) Method to find the Confidence Coefficient: Find the z value with area to its left, i.e., 1. Locate inside the Normal Probability Table (Table E) 2. Starting at , move your hand to the left along the row until you reach the Z column. This is the integer and tenths digits. Go back to , next move your hand to the top of its column. This is the hundreds digits. 3.

Add the integer and tenths digits to the hundredths digits to find the value for z.

4.

Affix a

sign in front of the number.

Dr. Janet Winter, [email protected]

Stat 200

Page 4

Using the method described, complete the table below. The first 2 have been completed for you (answer at the end). Confidence Level 1−α

α

.90 .95 .99 .80 .98 .96 .93

.10 .05

α/2

(1 − α ) +

.95 .975

α 2

.95 .975

Confidence Coefficient 𝑧(𝑎/2) 1.645 1.96

III. Development of the Confidence Interval Formula 𝜎 𝜎 𝑥̅ − 𝑧 < 𝜇 < 𝑥̅ + 𝑧 √𝑛 √𝑛

Whenever the population standard deviation 𝜎 is known and either the population is normally distributed or n ≥ 30, the Central Limit Theorem guarantees the sample mean is normally distributed or: 𝑥̅ − 𝜇 −(−𝜇) > −(−𝑥̅ + 𝑧𝜎𝑥̅ ) 𝑥̅ + 𝑧 ∙ 𝜎𝑥̅ > 𝜇 > 𝑥̅ − 𝑧𝜎𝑥̅ 𝑥̅ − 𝑧 ∙ 𝜎𝑥 < 𝜇 < 𝑥̅ + 𝑧𝜎𝑥̅ 𝑥̅ − 𝑧

𝜎

√𝑛

< 𝜇 < 𝑥̅ + 𝑧

𝜎

√𝑛

Note: If the population standard deviation is not known or stated, use x� − t

s

√n

< 𝜇 < x� + t

s

√n

Dr. Janet Winter, [email protected]

(see section E page 9).

Stat 200

Page 5

IV. Review of Concepts and Maximum Error of Estimate is the point estimate and the center of the confidence interval z is the confidence coefficient, the number of multiples of the standard error needed to construct an interval estimate of the correct width to have a level of confidence 1− α

is called the maximum error of estimate. V. Example: 35 fifth-graders have a mean reading score of 82. The standard deviation of the population is 15. a) Find the 95% confidence interval estimate for the mean reading scores of all fifthgraders. Since we know the population standard deviation and n≥30, use . Use Table E backwards with the area to the left of z equal to .025. The value of z or the confidence coefficient is z = 1.9 + .06 = 1.96. This means approximately 95% of the sample means will fall within 1.96 standard errors of the population mean. Use z = 1.96 in the formula.

4.97, rounded to 5, is the maximum error of estimate. Be sure to list it for full credit in your answers.

Dr. Janet Winter, [email protected]

Stat 200

Page 6

b) Find the 99% confidence interval estimate of the mean reading scores of all fifthgraders. Since approximately 99% of the sample means will fall within 2.58 standard errors of the population mean, use z = 2.58 𝑋� = 82.1, 𝑛 = 35, 𝜎 = 15 𝑋� − 𝑧

𝜎

√𝑛

< 𝜇 < 𝑋� + 𝑧

82.1 − 2.58

15

√35

𝜎

√𝑛

< 𝜇 < 82.1 + 2.58

82.1 − 6.54 < 𝜇 < 82.1 + 6.54

15

√35

82.1 ± 6.5

75.6 < 𝜇 < 88.5

6.54, rounded to 6.5, is the maximum error of estimate. Be sure to list it in the next to last step. c) Is the 95% confidence interval or the 99% confidence interval larger? Explain why. 95% confidence level: 77 < μ < 87 99% confidence level: 75 < μ < 89 The 99% confidence level is larger because it has a larger z value.

Question 1 A study of 40 English composition professors showed that they spent, on average, 12.6 minutes correcting a student’s term paper. Find the 90% confidence interval of the mean time for all composition papers when 𝜎 = 2.5 minutes. n = 40 𝑋�= 12.6

Since the population standard deviation is given and n = 40 is greater than 30, use the formula: 𝝈 𝝈 �−𝒛 �+𝒛

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