confidence intervals for the mean (small samples) 1 larson/farber 4th ed

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Confidence Intervals for the Mean (Small Samples) 1 Larson/Farber 4th ed

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Confidence Intervals for the Mean (Small Samples)

1Larson/Farber 4th ed

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 ed2

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 ed3

-xtsn

1. 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.

• d.f. = n – 1 Degrees of freedom

Larson/Farber 4th ed4

x

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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.

t0

Standard normal curve

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

d.f. = 2

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

Larson/Farber 4th ed6

Table 5: t-Distribution

tc = 2.145

Solution: d.f. = n – 1 = 15 – 1 = 14

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

Larson/Farber 4th ed7

t

-tc = -2.145 tc = 2.145

c = 0.95

A c-confidence interval for the population mean μ

The probability that the confidence interval contains μ is c.

Larson/Farber 4th ed8

where c

sx E x E E t

n

Larson/Farber 4th ed9

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.

xxn

2( )1

x xsn

cE tn

s

d.f. = n – 1

x

In Words In Symbols

Larson/Farber 4th ed10

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

Left endpoint: Right endpoint: Interval:

x Ex E

x E x E

In Words In Symbols

You randomly select 16 coffee shops and measure the temperature of the coffee sold at each. The sample mean temperature is 162.0ºF with a sample standard deviation of 10.0ºF. Find the 95% confidence interval for the mean temperature. Assume the temperatures are approximately normally distributed.

Larson/Farber 4th ed11

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

n =16, x = 162.0 s = 10.0 c = 0.95 df = n – 1 = 16 – 1 = 15 Critical Value

Larson/Farber 4th ed12

Table 5: t-Distribution

tc = 2.131

Margin of error:

Confidence interval:

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102.131 5.316cE t

n s

162 5.3

156.7

x E

162 5.3

167.3

x E

Left Endpoint:

Right Endpoint:

156.7 < μ < 167.3

156.7 < μ < 167.3

Larson/Farber 4th ed14

( )

• 162.0156.

7167.3

With 95% confidence, you can say that the mean temperature of coffee sold is between 156.7ºF and 167.3ºF.

Point estimate

xx E x E

No

Larson/Farber 4th ed15

Is n 30?

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

distribution or the t-distribution. Yes

Is known?

No

Use the normal distribution with

If is unknown, use s instead.

cE zn

σYes

No

Use the normal distribution with

cE zn

σ

Yes

Use the t-distribution with

and n – 1 degrees of freedom.

cE tn

s

You randomly select 25 newly constructed houses. The sample mean construction cost is $181,000 and the population standard deviation is $28,000. Assuming construction costs are normally distributed, should you use the normal distribution, the t-distribution, or neither to construct a 95% confidence interval for the population mean construction cost?

Larson/Farber 4th ed16

Solution:Use the normal distribution (the population is normally distributed and the population standard deviation is known)

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 ed17