1 / 17

Confidence Intervals for Mean with Small Samples: Interpretation and Construction

Learn how to interpret and use the t-distribution, construct confidence intervals, and determine critical values for the mean when sample size is less than 30 and population is normally distributed with unknown standard deviation.

baerd
Download Presentation

Confidence Intervals for Mean with Small Samples: Interpretation and Construction

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. Section 6.2 Confidence Intervals for the Mean (Small Samples) Larson/Farber 4th ed

  2. 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

  3. 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

  4. 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 Larson/Farber 4th ed

  5. d.f. = 2 d.f. = 5 Properties of 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. t 0 Standard normal curve Larson/Farber 4th ed

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

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

  8. Confidence Intervals for the Population Mean A c-confidence interval for the population mean μ • The probability that the confidence interval contains μis c. Larson/Farber 4th ed

  9. Confidence Intervals and t-Distributions 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. d.f. = n – 1 Larson/Farber 4th ed

  10. Confidence Intervals and t-Distributions In Words In Symbols Left endpoint: Right endpoint: Interval: Find the left and right endpoints and form the confidence interval. Larson/Farber 4th ed

  11. Example: Constructing a Confidence Interval 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. Solution: Use the t-distribution (n < 30, σ is unknown, temperatures are approximately distributed.) Larson/Farber 4th ed

  12. Solution: Constructing a Confidence Interval • n =16, x = 162.0 s = 10.0 c = 0.95 • df = n – 1 = 16 – 1 = 15 • Critical Value Table 5: t-Distribution tc = 2.131 Larson/Farber 4th ed

  13. Solution: Constructing a Confidence Interval • Margin of error: • Confidence interval: Left Endpoint: Right Endpoint: 156.7 < μ < 167.3 Larson/Farber 4th ed

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

  15. 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? Cannot use the normal distribution or the t-distribution. Is  known? Larson/Farber 4th ed

  16. Example: Normal or t-Distribution? 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? Solution: Use the normal distribution (the population is normally distributed and the population standard deviation is known) Larson/Farber 4th ed

  17. 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

More Related