Chapter 8

1. Chapter 8 Confidence Intervals

2. Chapter Outline 8.1 z-Based Confidence Intervals for a Population Mean: σ Known 8.2 t-Based Confidence Intervals for a Population Mean: σ Unknown 8.3 Sample Size Determination 8.4 Confidence Intervals for a Population Proportion 8.5 A Comparison of Confidence Intervals and Tolerance Intervals (Optional)

3. 8.1 z-Based Confidence Intervals for a Mean: σ Known • If a population is normally distributed with mean μ and standard deviation σ, then the sampling distribution of x is normal with mean μx= μ and standard deviation • Use a normal curve as a model of the sampling distribution of the sample mean • Exactly, because the population is normal • Approximately, by the Central Limit Theorem for large samples

4. The Empirical Rule • 68.26% of all possible sample means are within one standard deviation of the population mean • 95.44% of all possible observed values of x are within two standard deviations of the population mean • 99.73% of all possible observed values of x are within three standard deviations of the population mean

5. Example 8.1 The Car Mileage Case • Assume a sample size (n) of 5 • Assume the population of all individual car mileages is normally distributed • Assume the standard deviation (σ) is 0.8 • The probability that x with be within ±0.7 (2σx≈0.7) of µ is 0.9544

6. The Car Mileage Case Continued • Assume the sample mean is 31.3 • That gives us an interval of [31.3 ± 0.7] = [30.6, 32.0] • The probability is 0.9544 that the interval [x ± 2σ] contains the population mean µ

7. Generalizing • In the example, we found the probability that m is contained in an interval of integer multiples of sx • More usual to specify the (integer) probability and find the corresponding number of sx • The probability that the confidence interval will not contain the population mean m is denoted by • In the mileage example, a = 0.0456

8. Generalizing Continued • The probability that the confidence interval will contain the population mean μ is denoted by 1 -  • 1 –  is referred to as the confidence coefficient • (1 – )  100% is called the confidence level • Usual to use two decimal point probabilities for 1 –  • Here, focus on 1 –  = 0.95 or 0.99

9. General Confidence Interval • In general, the probability is 1 – a that the population mean m is contained in the interval • The normal point za/2 gives a right hand tail area under the standard normal curve equal to a/2 • The normal point - za/2 gives a left hand tail area under the standard normal curve equal to a/2 • The area under the standard normal curve between -za/2 and za/2 is 1 – a

10. Sampling Distribution Of All Possible Sample Means Figure 8.2

11. z-Based Confidence Intervals for a Mean with σ Known

12. 95% Confidence Interval Figure 8.3

13. 99% Confidence Interval Figure 8.4

14. The Effect of a on Confidence Interval Width za/2 = z0.025 = 1.96 za/2 = z0.005 = 2.575 Figures 8.2 to 8.4

15. 8.2 t-Based Confidence Intervals for a Mean: σ Unknown • If σ is unknown (which is usually the case), we can construct a confidence interval for m based on the sampling distribution of • If the population is normal, then for any sample size n, this sampling distribution is called the t distribution

16. The t Distribution • Symmetrical and bell-shaped • The t distribution is more spread out than the standard normal distribution • The spread of the t is given by the number of degrees of freedom (df) • For a sample of size n, there are one fewer degrees of freedom, that is, df = n – 1

17. Degrees of Freedom and thet-Distribution Figure 8.6

18. The t Distribution and Degrees of Freedom • As the sample size n increases, the degrees of freedom also increases • As the degrees of freedom increase, the spread of the t curve decreases • As the degrees of freedom increases indefinitely, the t curve approaches the standard normal curve • If n ≥ 30, so df = n – 1 ≥ 29, the t curve is very similar to the standard normal curve

19. t and Right Hand Tail Areas • t is the point on the horizontal axis under the t curve that gives a right hand tail equal to  • So the value of t in a particular situation depends on the right hand tail area  and the number of degrees of freedom

20. t and Right Hand Tail Areas Figure 8.7

21. Using the t Distribution Table Table 8.3 (top)

22. t-Based Confidence Intervals for a Mean: σ Unknown Figure 8.10

23. 8.3 Sample Size Determination If σ is known, then a sample of sizeso that x is within B units of , with 100(1-)% confidence

24. Example (Car Mileage Case)

25. Sample Size Determination (t) If σ is unknown and is estimated from s, then a sample of sizeso that x is within B units of , with 100(1-)% confidence. The number of degrees of freedom for the ta/2 point is the size of the preliminary sample minus 1

26. Example 8.6: The Car Mileage Case

27. 8.4 Confidence Intervals for a Population Proportion • If the sample size n is large, then a (1­a)100% confidence interval for ρ is • Here, n should be considered large if both • n · p̂ ≥ 5 • n · (1 – p̂) ≥ 5

28. Example 8.8: The Cheese Spread Case

29. Determining Sample Size for Confidence Interval for ρ • A sample size given by the formula…will yield an estimate p̂, precisely within B units of ρ, with 100(1 - )% confidence • Note that the formula requires a preliminary estimate of p. The conservative value of p = 0.5 is generally used when there is no prior information on p

30. 8.5 A Comparison of Confidence Intervals and Tolerance Intervals (Optional) • A tolerance interval contains a specified percentage of individual population measurements • Often 68.26%, 95.44%, 99.73% • A confidence interval is an interval that contains the population mean μ, and the confidence level expresses how sure we are that this interval contains μ • Confidence level set high (95% or 99%) • Such a level is considered high enough to provide convincing evidence about the value of μ

31. Selecting an Appropriate Confidence Interval for a Population Mean Figure 8.19