Ch4 Confidence Interval
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This document discusses key concepts in statistics for engineers and scientists such as point estimates, properties of good estimators, confidence intervals, and the t-distribution. A point estimate is a single numerical value used to estimate a population parameter from a sample. A good estimator must be unbiased, consistent, and relatively efficient. A confidence interval provides a range of values that is likely to contain the true population parameter based on the sample data and confidence level. The t-distribution is similar to the normal distribution but has greater variance and depends on degrees of freedom. Examples are provided to demonstrate how to calculate confidence intervals for means using the normal and t-distributions.
Ch4 Confidence Interval
- 1. STATISTICS FOR ENGINEERS AND SCIENTISTS Farhan Alfin Confidence Interval
- 2. point estimate • A point estimate is a single numerical value, computed from sample information, used to estimate a given population parameter. • The sample mean is a point estimate for the population mean.
- 3. Properties of a Good Estimator The estimator must be an unbiased estimator. That is, the expected value or the mean of the estimates obtained from samples of a given size is equal to the parameter being estimated.
- 4. Properties of a Good Estimator The estimator must be consistent. For a consistent estimator, as sample size increases, the value of the estimator approaches the value of the parameter estimated.
- 5. Properties of a Good Estimator The estimator must be a relatively efficient estimator. That is, of all the statistics that can be used to estimate a parameter, the relatively efficient estimator has the smallest variance.
- 6. Confidence Intervals An interval estimate of a parameter is an interval or a range of values used to estimate the parameter. This estimate may or may not contain the value of the parameter being estimated.
- 7. Confidence Intervals A confidence interval is a specific interval estimate of a parameter determined by using data obtained from a sample and the specific confidence level of the estimate.
- 8. Confidence Intervals Confidence Level The confidence level of an interval estimate of a parameter is the probability that the interval estimate will contain the parameter.
- 9. Confidence Intervals Maximum Error of Estimate The maximum error of estimate is the maximum difference between the point estimate of a parameter and the actual value of the parameter.
- 10. Confidence Intervals Maximum Error of Estimate • Now, to find a confidence interval estimate , we need to compute the standard deviation for the sampling distribution of the sample mean.
- 11. Confidence Intervals for the Mean ( Known or n 30) and Sample Size 2 2 X z X z n n The confidence level is the percentage equivalent to the decimal value of 1 – .
- 12. • The following table lists values for z and z/2 when = 0.1, 0.05, and 0.01. • These values can be obtained from the standard normal tables.
- 13. Notes : • Note: The notation z/2 z/2. • First, we have to divide by 2, then you find the corresponding z score value. • The relationship between and the confidence level is that the stated confidence level is the percentage equivalent to the decimal value of 1 - . • For example, if we are constructing a 98% confidence interval, then 0.98 = 1 - , from which = 0.02.
- 14. Confidence Intervals - Example The president of wheat research Center wishes to estimate the average protein contain of the new harvest. From past studies, the standard deviation is known to be 1.1%. A sample of 50 wheat is selected, and the mean is found to be 11%. Find the 95% confidence interval of the population mean.
- 15. Confidence Intervals - Example Since the 95% confidence interval is desired, Zα/2 =1.96, Hence, Substituting in the formula 2 2 X z X z n n 1.1 11 11 1.96 11 1.96 50 50 11 0.3 11 0.3 10.7 11.3 or 11 0.3
- 16. Characteristics of the t Distribution The t distribution shares some characteristics of the normal distribution and differs from it in others. The t distribution is similar to the standard normal distribution in the following ways: It is bell-shaped. It is symmetrical about the mean.
- 17. Characteristics of the t Distribution The mean, median, and mode are equal to 0 and are located at the center of the distribution. The curve never touches the x axis. The t distribution differs from the standard normal distribution in the following ways:
- 18. Characteristics of the t Distribution The variance is greater than 1. The t distribution is actually a family of curves based on the concept of degrees of freedom, which is related to the sample size. As the sample size increases, the t distribution approaches the standard normal distribution.
- 19. Standard Normal Curve and the t Distribution
- 20. Confidence Interval for the Mean ( Unknown and n < 30) - Example Ten randomly selected bread loaf, and the weight of the loaf was measured. The mean was 90 gr, and the standard deviation was 2 gr. Find the 95% confidence interval of the mean weigh. Assume that the variable is approximately normally distributed.
- 21. Confidence Interval for the Mean ( Unknown and n < 30) - Example Since is unknown and s must replace it, the t distribution must be used with = 0.05. Hence, with 9 degrees of freedom, t/2 = 2.262.
- 22. Confidence Interval for the Mean ( Unknown and n < 30) - Example 2 2 5 5 90 2.262 90 2.262 10 10 90 3.58 90 3.58 86.42 93.58 or 90 3.58 s s X t X t n n