Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Chapter 6 Confidence Intervals § 6.2 Confidence Intervals for the Mean (Small Samples) The t-Distribution When a sample size is less than 30, and the random variable x is approximately normally distributed, it follow a t-distribution. t x μ s n 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 x 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 Continued. Larson & Farber, Elementary Statistics: Picturing the World, 3e 3 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. The tails in the t-distribution are “thicker” than those in the standard normal distribution. d.f. = 2 d.f. = 5 0 t Standard normal curve Larson & Farber, Elementary Statistics: Picturing the World, 3e 4 Critical Values of t Example: Find the critical value tc for a 95% confidence when the sample size is 5. Appendix B: Table 5: t-Distribution Level of confidence, c One tail, d.f. Two tails, 1 2 3 4 5 0.50 0.25 0.50 1.000 .816 .765 .741 .727 0.80 0.10 0.20 3.078 1.886 1.638 1.533 1.476 0.90 0.95 0.98 0.05 0.025 0.01 0.10 0.05 0.02 6.314 12.706 31.821 2.920 4.303 6.965 2.353 3.182 4.541 2.132 2.776 3.747 2.015 2.571 3.365 d.f. = n – 1 = 5 – 1 = 4 tc = 2.776 c = 0.95 Larson & Farber, Elementary Statistics: Picturing the World, 3e Continued. 5 Critical Values of t Example continued: Find the critical value tc for a 95% confidence when the sample size is 5. 95% of the area under the t-distribution curve with 4 degrees of freedom lies between t = ±2.776. c = 0.95 tc = 2.776 tc = 2.776 Larson & Farber, Elementary Statistics: Picturing the World, 3e t 6 Confidence Intervals and t-Distributions Constructing a Confidence Interval for the Mean: Distribution In Words t- In Symbols 1. Identify the sample statistics n, x, 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. 4. Find the left and right endpoints and form the confidence interval. x x n ( x x )2 s n 1 d.f. = n – 1 E tc s n Left endpoint: x E Right endpoint: x E Interval: x E x E Larson & Farber, Elementary Statistics: Picturing the World, 3e 7 Constructing a Confidence Interval Example: In a random sample of 20 customers at a local fast food restaurant, the mean waiting time to order is 95 seconds, and the standard deviation is 21 seconds. Assume the wait times are normally distributed and construct a 90% confidence interval for the mean wait time of all customers. n = 20 x 95 d.f. = 19 tc = 1.729 x E = 95 ± 8.1 s = 21 21 8.1 E tc s 1.729 n 20 86.9 < μ < 103.1 We are 90% confident that the mean wait time for all customers is between 86.9 and 103.1 seconds. Larson & Farber, Elementary Statistics: Picturing the World, 3e 8 Normal or t-Distribution? Use the normal distribution with Is n 30? Yes E zc σ . n If is unknown, use s instead. No Is the population normally, or approximately normally, distributed? No Yes You cannot use the normal distribution or the t-distribution. Use the normal distribution with Is known? No Yes E zc σ . n Use the t-distribution with E tc s n and n – 1 degrees of freedom. Larson & Farber, Elementary Statistics: Picturing the World, 3e 9 Normal or t-Distribution? Example: Determine whether to use the normal distribution, the t-distribution, or neither. a.) n = 50, the distribution is skewed, s = 2.5 The normal distribution would be used because the sample size is 50. b.) n = 25, the distribution is skewed, s = 52.9 Neither distribution would be used because n < 30 and the distribution is skewed. c.) n = 25, the distribution is normal, = 4.12 The normal distribution would be used because although n < 30, the population standard deviation is known. Larson & Farber, Elementary Statistics: Picturing the World, 3e 10