Percentage point z /2

Transcription

1 Chapter 8: Statistical Intervals Why? point estimate is not reliable under resampling. Interval Estimates: Bounds that represent an interval of plausible values for a parameter There are three types of intervals: Confidence intervals focusing on this Prediction intervals Tolerance intervals Confidence Level : between 0 and 1 A confidence level: 1 - or 100(1- )%. E.g. 95%. This is the proportion of times that the confidence interval actually does contain the population parameter, assuming that the estimation process is repeated a large number of times. Other names: degree of confidence or the confidence coefficient. Percentage point z /2 Given Finding z /2 for 95% Degree of Confidence = 5% /2 = 2.5% =.025 III Percentage point 8-1 Confidence Interval on the Mean of a Normal Distribution, Variance Known Let X 1,, X n be a random sample from X~N(, 2 ) with the sample mean X. We have Confidence Interval on the Mean of a Normal Distribution, Variance Known X Z ~ N(0,1) n X Z ~ N(0,1) n 1

2 8-1.1 Confidence Interval on the Mean of a Normal Distribution, Variance Known Example 8-1 Recap: Interpreting a Confidence Interval Confidence Level and Precision of Error The confidence interval is a random interval The length of a confidence interval is a measure of the precision of estimation= 2z /2 n Figure 8-2 Error in estimating with x. The half length of CI Margin of error Choice of Sample Size Choice of Sample Size: Example

3 8-1.3 One-Sided Confidence Bounds A Large-Sample Confidence Interval for Central Limit Theorem implies that Example 8-3: Read If the sample size is small, and the population distribution is normal, then use t-value instead of z-value Example 8-5 Mercury Contamination Example 8-5 (continued) Example 8-5 (continued) 8-2 Confidence Interval on the Mean of a The t distribution Student in

4 8-2 Confidence Interval on the Mean of a The t distribution 8-2 Confidence Interval on the Mean of a The t-value T(df) Z as df Figure 8-4 Probability density functions of several t distributions. Figure 8-5 Percentage points of the t distribution. Table V The t Confidence Interval on Example One-sided confidence bounds on the mean are found by replacing t /2,n-1 in Equation 8-16 with t,n Confidence Interval on the Mean of a 8-3 Confidence Interval on the Mean of a Figure 8-6 Box and Whisker plot for the load at failure data in Example 8-6. Figure 8-7 Normal probability plot of the load at failure data in Example

5 (Skip) Definition Figure 8-8 Probability density functions of several 2 distributions. Definition Percentage points Table IV One-Sided Confidence Bounds Example 8-7 5

6 8-4 A Large-Sample Confidence Interval For a Population Proportion 8-4 A Large-Sample Confidence Interval For a Population Proportion Recall: Normal Approximation for Binomial Proportion The quantity p ( 1 p) / n is called the standard error of the point estimator ˆ X P. n CI: (point estimate) (critical value)*(standard error) (point estimate) (margin of error) 8-4 Example 8-8 (proportion) Cranshaft Bearing 8-4 Choice of Sample Size (proportion) The sample size for a specified value E is given by Example 8-9 (Proportion) 8-4 One-Sided Confidence Bounds (Proportion) (skip) 6

7 8.5 Guidelines for Constructing Confidence Intervals 8-7 Tolerance and Prediction Intervals Prediction Interval for Future Observation 8 The prediction interval for X n+1 will always be longer than the confidence interval for. Prediction error: X n+1 X Mean of prediction error: 0 Variance of prediction error: V(X n+1 X)=σ 2 + σ2 n = σ2 (1+1/n) 8-7 Example 8-11 Alloy Adhesion Tolerance Interval for a Normal Distribution 8-7 Example