section 6.2 confidence intervals for the mean (small samples) larson/farber 4th ed
DESCRIPTION
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 t c. Larson/Farber 4th edTRANSCRIPT
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