Confidence Intervals & Sample Size

Chapter 7

Introduction

• One aspect of inferential statistics is estimation, which is the process of estimating the value of a parameter from information obtained from a sample • One important question exists: • How large should the sample be in order to make an accurate estimate?

•

7.1 – Confidence Intervals for the Mean When σ is Known

Point estimate:

• Specific numerical value estimate of a parameter • The best point estimate of a population mean μ is the sample mean 1.

2.

3.

Three Properties of a Good Estimator

The estimator should be an unbiased estimator The estimator should be consistent estimator The estimator should be a relatively efficient estimator

Interval Estimates

•

Interval estimate

• Interval or a range of values used to estimate the parameter • This estimate may or may not contain the value of the parameter being estimated •

Confidence level

• Probability that the interval estimate will contain the parameter, assuming that a large number of samples are selected and that the estimation process on the same parameter is repeated •

Confidence interval

• Determined by using data obtained from a sample and by using the specific confidence level of the estimate

Confidence Interval Formula

• Three common confidence levels exist: 90, 95, and 99%

Formula for the Confidence Interval of the Mean

𝛼 2 𝜎 𝑛 𝛼 2 𝜎 𝑛 •

Maximum error of the mean:

• Maximum likely difference between the point estimate of a parameter and the actual value of the parameter • 𝑧 𝛼 2 𝜎 of error) 𝑛 is the maximum error of the mean (also called margin •

Rounding rule:

• Round off to one more decimal place than original data

Examples

• 7 – 1 • A researcher wishes to estimate the number of days it takes an automobile dealer to sell a Chevrolet Aveo. A sample of 50 cars had a mean time on the dealer’s lot of 54 days. Assume the population standard deviation to be 6.0 days. Find the best point estimate of the population mean and the 95% confidence interval of the population mean.

Example

• 7 – 2 • A survey of 30 adults found that the mean age of a person’s primary vehicle is 5.6 years. Assuming the standard deviation of the population is 0.8 year, find the best point estimate of the population mean and the 99% confidence interval of the population mean.

Examples

• 7 – 3 • The following data represent a sample of the assets (in millions of dollars) of 30 credit unions in southwestern Pennsylvania. Find the 90% confidence interval of the mean. (use calculators) 12.23

16.56

4.39

2.89

13.19

7.92

1.24

3.17

12.77

1.06

12.24

73.25

40.22

9.16

4.78

2.17

18.13

2.76

11.59

5.01

1.91

2.42

1.42

16.85

8.74

2.27

6.69

1.47

14.64

21.58

Sample Size

• One important question exists… • How large a sample is necessary to make an accurate estimate?

Formula for the Minimum Sample Size Needed for an Interval Estimate of the Population Mean

𝑛 = 𝑧 𝛼 2 ∗ 𝜎 2 𝐸 • Where E is the maximum error of the estimate. If necessary, round the answer up to obtain a whole number.

Examples

• 7 – 4 • A scientist wishes to estimate the average depth of a river. He wants to be 99% confident that the estimate is accurate within 2 feet. From a previous study, the standard deviation of the depths measured was 4.38

•

7.2 – Confidence Intervals when σ is Unknown

Most of the time the population standard deviation (σ) is not known • It is estimated using the sample standard deviation (s) • Values are taken from the t distribution • The t distribution was formulated in 1908 by an Irish brewing employee named W.S. Gosset

•

Characteristics of the t Distribution

The t distribution shares some characteristics of the normal distribution and differs it in others.

The t distribution is similar to the normal distribution in these ways:

1.

Bell-shaped 2.

3.

Symmetric about the mean Mean, median, and mode are equal to 0 and are located at the center of the distribution 4.

Curve never touches the x axis

The t distribution differs from the normal distribution in these ways:

1.

Variance is greater than 1 2.

3.

t distribution is a family of curves based on concept of degrees of freedom which is related to sample size As sample size increases, t distribution approaches normal distribution

Degrees of Freedom

•

Degrees of freedom

• Number of values that are free to vary after a sample statistic has been computed • Tell the researcher which specific curve to use • • The symbol d.f. is used for degrees of freedom Degrees of freedom for a confidence interval for the mean are found by subtracting 1 from the sample size: d.f. = n – 1 • • When d.f. is greater than 30, it may fall between two table values Always round down to the nearest table value • Example: 68 is closer to 70, but round down to table value of 65

Formula

Formula for a Specific Confidence Interval for the Mean When σ is Unknown and n < 30

𝑋 − 𝑡

𝛼 2

𝑠 𝑛

𝛼 2

𝑠 𝑛

•

The degrees of freedom are n – 1

Examples

• 7 – 5 • Find the t

a/2

size is 22 value for a 95% confidence interval when the sample • 7 – 6 • Ten randomly selected people were asked how long they slept at night. The mean time was 7.1 hours, and the standard deviation was 0.78 hour. Find the 95% confidence interval of the mean time. Assume the variable is normally distributed.

Example 7 – 7

• The data represent a sample of the number of home fires started by candles for the past several years. Find the 99% confidence interval for the mean number of home fires started by candles each year.

5460 5900 6090 6310 7160 8440 9930

•

7.3 – Confidence Intervals and Sample Size for Proportions

Proportion

• Represents a part of a whole expressed as fraction, decimal, or percentage • Proportions can also represent probabilities

Symbols Used in Proportion Notation

𝑝 = 𝑝𝑜𝑝𝑢𝑙𝑎𝑡𝑖𝑜𝑛 𝑝𝑟𝑜𝑝𝑜𝑟𝑡𝑖𝑜𝑛 𝑝 𝑟𝑒𝑎𝑑 p hat = 𝑠𝑎𝑚𝑝𝑙𝑒 𝑝𝑟𝑜𝑝𝑜𝑟𝑡𝑖𝑜𝑛 For a sample proportion: 𝑋 𝑝 = 𝑛 and 𝑛−𝑋 or 𝑛 where X = number of sample units that posses the characteristics of interest and n = sample size

Example 7 – 8

• In a recent survey of 150 households, 54 had central air conditioning. Find p hat and q hat where p hat is the proportion of households that have central air conditioning.

Formula

• Maximum error of the estimate is 𝐸 = 𝑧 𝛼 2 𝑛 • Confidence intervals about proportions must meet the criteria that 𝑛 𝑝 ≥ 5 𝑎𝑛𝑑 𝑛

Formula for a Specific Confidence Interval for a Proportion

𝑝 − 𝑧 𝛼 2 𝑛 < 𝑝 < 𝑝 + 𝑧 𝛼 2 𝑛

Examples

• 7 – 9 • A sample of 500 nursing applications included 60 from men. Find the 90% confidence interval of the true proportion of men who applied to the nursing program.

• 7 – 10 • A survey of 1721 people found that 15.9% of individuals purchase religious books at a Christian bookstore. Find the 95% confidence interval of the true proportion of people who purchase their religious books at a Christian bookstore.

Sample Size for Proportions

• To find the sample size needed to determine a confidence interval about a proportion use:

Formula for Minimum Sample Size Needed for Interval Estimate of a Population Proportion

𝑧 𝛼 2 𝐸 2 • If necessary, round up to obtain a whole number

Examples

• 7 – 11 • A researcher wishes to estimate, with 95% confidence, the proportion of people who own a home computer. A previous study shows that 40% of those interviewed had a computer at home. The researcher wishes to be accurate within 2% of the true proportion. Find the minimum same size necessary.

• 7 – 12 • The same researcher wishes to estimate the proportion of executives who own a car phone. She wants to be 90% confident and be accurate within 5% of the true proportion. Find the minimum sample size necessary.

• •

7.4 – Confidence Intervals for Variances & Stand. Deviations

In statistics, the variance and standard deviation of a variable are as important as the mean For example: • Products that fit together (such as pipes) are manufactured so that variations in diameters are as small as possible

Chi-Square Distribution

• To calculate these confidence intervals, a new statistical distribution is needed •

Chi-square distribution

• • Similar to the t variable in that its distribution is a family of curves based of the number of degrees of freedom Symbol for chi-square is 𝜒 2 • • Chi-square variable cannot be negative Area under the chi-square distribution is 1.00 or 100%

Example 7 – 13

• Find the values of 𝜒 2 interval when n = 25 𝑟𝑖𝑔ℎ𝑡 and 𝜒 2 𝑙𝑒𝑓𝑡 for a 90% confidence

Formulas

• Formulas for confidence intervals where d.f. = n – 1

Formula for the Confidence Interval for a Variance

(𝑛 − 1)𝑠 2 < 𝜎 2 𝜒 𝑟𝑖𝑔ℎ𝑡 2 < (𝑛 − 1)𝑠 2 𝜒 𝑙𝑒𝑓𝑡 2

Formula for the Confidence Interval for a Standard Deviation

(𝑛 − 1)𝑠 2 < 𝜎 < 𝜒 𝑟𝑖𝑔ℎ𝑡 2 (𝑛 − 1)𝑠 2 𝜒 𝑙𝑒𝑓𝑡 2

Examples

• 7 – 14 • Find the 95% confidence interval for the variance and standard deviation of the nicotine content of cigarettes manufactured if a sample of 20 cigarettes has a standard deviation of 1.6 milligrams.

• 7 – 15 • Find the 90% confidence interval for the variance and standard deviation for the price in dollars of an adult single-day ski lift ticket. The data represent a selected sample of nationwide ski resorts. Assume the variable is normally distributed.

59 39 54 49 53 46 52 49 51 48