Section 6.2

Confidence Intervals for the Mean (Small Samples)

Larson/Farber 4th ed

Section 6.2 Objectives

• Interpret the t-distribution and use a t-distribution table

• Construct confidence intervals when n < 30, the population is normally distributed, and σ is unknown

Larson/Farber 4th ed

The t-Distribution

• When the population standard deviation is unknown, the sample size is less than 30, and the random variable x is approximately normally distributed, it follows a t-distribution.

• Critical values of t are denoted by tc.

Larson/Farber 4th ed

Properties of the t-Distribution1. The t-distribution is bell shaped and symmetric

about the mean.2. The t-distribution is a family of curves, each

determined by a parameter called the degrees of freedom. The degrees of freedom are the number of free choices left after a sample statistic such as is calculated. When you use a t-distribution to estimate a population mean, the degrees of freedom are equal to one less than the sample size.

1. d.f. = n – 1 Degrees of freedom

Larson/Farber 4th ed

Properties of the t-Distribution

3. The total area under a t-curve is 1 or 100%.4. The mean, median, and mode of the t-distribution are

equal to zero.5. As the degrees of freedom increase, the t-distribution

approaches the normal distribution. After 30 d.f., the t-distribution is very close to the standard normal z-distribution.

t0Standard normal curve

The tails in the t-distribution are “thicker” than those in the standard normal distribution.d.f. = 5

d.f. = 2

Larson/Farber 4th ed

Example: Critical Values of t

Find the critical value tc for a 95% confidence when the sample size is 15.

Table 5: t-Distribution

tc = 2.145

Solution: d.f. = n – 1 = 15 – 1 = 14

Larson/Farber 4th ed

Solution: Critical Values of t

95% of the area under the t-distribution curve with 14 degrees of freedom lies between t = +2.145.

t

-tc = -2.145 tc = 2.145

c = 0.95

Larson/Farber 4th ed

Confidence Intervals for the Population Mean

A c-confidence interval for the population mean μ •

• The probability that the confidence interval contains μ is c.

Larson/Farber 4th ed

Confidence Intervals and t-Distributions

1. Identify the sample statistics n, , and s.

2. Identify the degrees of freedom, the level of confidence c, and the critical value tc.

3. Find the margin of error E.

d.f. = n – 1

Larson/Farber 4th ed

In Words In Symbols

Confidence Intervals and t-Distributions

4. Find the left and right endpoints and form the confidence interval.

Left endpoint: Right endpoint: Interval:

Larson/Farber 4th ed

In Words In Symbols

Example: Constructing a Confidence Interval

In a random sample of seven computers, the mean repair cost was $100 and the standard deviation was $42.50 - assume normal distribution. Construct a 95% confidence interval for the population mean.

Solution:Use the t-distribution (n < 30, σ is unknown, repairs are normally distributed.)

Larson/Farber 4th ed

Solution: Constructing a Confidence Interval

• n =7, x = $100.00 s = $42.50 c = 0.95• df = n – 1 = 7 – 1 = 6• Critical Value tc = 2.447

Solution: Constructing a Confidence Interval

• Margin of error:

Left Endpoint: Right Endpoint:

$60.69 < μ

• Confidence interval:

< $139.31

Solution: Constructing a Confidence Interval

• 60.69 < μ < 139.31

( )• $100$60.69 139.31

With 95% confidence, you can say that the mean cost of repair is between $60.69 and $139.31.

Point estimate

No

Normal or t-Distribution?

Is n ≥ 30?

Is the population normally, or approximately normally, distributed? Cannot use the normal

distribution or the t-distribution. Yes

Is σ known?No

Use the normal distribution with If σ is unknown, use s instead.

Yes

No

Use the normal distribution with

Yes

Use the t-distribution with

and n – 1 degrees of freedom.Larson/Farber 4th ed

Example: Normal or t-Distribution?

In a random sample of 18 one person tents, the mean price was $144.19 and the standard deviation was $61.32. Assume the prices are normally distributed.

Solution:Use the the t-distribution (n < 30, the population is normally distributed and the population standard deviation is unknown)

Section 6.2 Summary

• Interpreted the t-distribution and used a t-distribution table

• Constructed confidence intervals when n < 30, the population is normally distributed, and σ is unknown

Larson/Farber 4th ed