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Understanding Confidence Intervals: From z-based to t-based Methods

This chapter provides a comprehensive overview of confidence intervals in statistics, covering z-based confidence intervals for population means with known standard deviations and t-based intervals for unknown standard deviations. It elucidates the importance of sample size determination and confidence intervals for population proportions. The chapter also highlights the differences between confidence intervals and tolerance intervals. Key principles such as the Central Limit Theorem and the implications of the degrees of freedom in t-distribution are discussed, all designed to enhance the understanding of statistical estimation.

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Understanding Confidence Intervals: From z-based to t-based Methods

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  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

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