html5-img
1 / 10

Confidence Intervals

Chapter 6. Confidence Intervals. Confidence Intervals for the Mean (Small Samples). § 6.2. 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. Properties of the t -distribution.

obert
Télécharger la présentation

Confidence Intervals

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Chapter 6 Confidence Intervals

  2. Confidence Intervals for the Mean (Small Samples) § 6.2

  3. 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. Properties of the t-distribution • The t-distribution is bell shaped and symmetric about the mean. • 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 Continued.

  4. d.f. = 2 d.f. = 5 t Standard normal curve 0 The t-Distribution • The total area under a t-curve is 1 or 100%. • The mean, median, and mode of the t-distribution are equal to zero. • 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.

  5. 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 d.f. = n – 1 = 5 – 1 = 4 tc = 2.776 c = 0.95 Continued.

  6. c = 0.95 t tc =  2.776 tc = 2.776 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.

  7. Confidence Intervals and t-Distributions Constructing a Confidence Interval for the Mean: t-Distribution In Words In Symbols • Identify the sample statistics n, , and s. • Identify the degrees of freedom, the level of confidence c, and the critical value tc. • Find the margin of error E. • Find the left and right endpoints and form the confidence interval. d.f. = n – 1 Left endpoint: ERight endpoint: +E Interval: E < μ< +E

  8. 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 = 95 s = 21 d.f.= 19 tc = 1.729 86.9 < μ< 103.1  ± E = 95 ± 8.1 We are 90% confident that the mean wait time for all customers is between 86.9 and 103.1 seconds.

  9. Use the normal distribution with If  is unknown, use s instead. Yes Yes No No Yes Use the normal distribution with No Use the t-distribution with and n – 1 degrees of freedom. Normal or t-Distribution? Is n  30? Is the population normally, or approximately normally, distributed? You cannot use the normal distribution or the t-distribution. Is  known?

  10. 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.

More Related