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

2.  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 2

3.  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 3

4. 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 4

5. 5 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

6. Find the critical value t c for a 95% confidence when the sample size is 15. Larson/Farber 4th ed 6 Table 5: t-Distribution t c = 2.145 Solution: d.f. = n – 1 = 15 – 1 = 14

7. 95% of the area under the t-distribution curve with 14 degrees of freedom lies between t = +2.145. Larson/Farber 4th ed 7 t -t c = -2.145 t c = 2.145 c = 0.95

8. A c-confidence interval for the population mean μ   The probability that the confidence interval contains μ is c. Larson/Farber 4th ed 8

9. 9 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 In WordsIn Symbols

10. Larson/Farber 4th ed 10 4.Find the left and right endpoints and form the confidence interval. Left endpoint: Right endpoint: Interval: In WordsIn Symbols

11. 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 ed 11 Solution: Use the t-distribution (n < 30, σ is unknown, temperatures are approximately distributed.)

12.  n =16, x = 162.0 s = 10.0 c = 0.95  df = n – 1 = 16 – 1 = 15  Critical Value Larson/Farber 4th ed 12 Table 5: t-Distribution t c = 2.131

13.  Margin of error:  Confidence interval: Larson/Farber 4th ed 13 Left Endpoint:Right Endpoint: 156.7 < μ < 167.3

14.  156.7 < μ < 167.3 Larson/Farber 4th ed 14 ( ) 162.0 156.7167.3 With 95% confidence, you can say that the mean temperature of coffee sold is between 156.7 ºF and 167.3 ºF. Point estimate

15. No Larson/Farber 4th ed 15 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.

16. 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 ed 16 Solution: Use the normal distribution (t he population is normally distributed and the population standard deviation is known)

17.  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 17