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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.

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Confidence Intervals for Mean with Small Samples: Interpretation and Construction

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  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

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