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 t c. Larson/Farber 4th ed

Properties of the t-Distribution 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 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. t 0 Standard 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 t c for a 95% confidence when the sample size is 15. Table 5: t-Distribution t c = 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 = t -t c = t c = 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 t c. 3.Find the margin of error E. d.f. = n – 1 Larson/Farber 4th ed In WordsIn 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 WordsIn Symbols

Example: Constructing a Confidence Interval In a random sample of seven computers, the mean repair cost was $100 and the standard deviation was $ 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 = $ s = $42.50 c = 0.95 df = n – 1 = 7 – 1 = 6 Critical Value t c = 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 < μ < ( ) $100$ With 95% confidence, you can say that the mean cost of repair is between $60.69 and $ 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  known? No Use the normal distribution with If  is unknown, use s instead. YesNo 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 $ and the standard deviation was $ 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