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#### Presentation Transcript

**1. **1 Introduction to Estimation Chapter 10

**2. **2 10.1 Introduction Statistical inference is the process by which we acquire information about populations from samples.
There are two types of inference:
Estimation
Hypotheses testing

**3. **3 10.2 Concepts of Estimation The objective of estimation is to determine the value of a population parameter on the basis of a sample statistic.
There are two types of estimators:
Point Estimator
Interval estimator

**4. **4 Point Estimator

**5. **5 Point Estimator

**6. **6 Interval Estimator

**7. **7 Selecting the right sample statistic to estimate a parameter value depends on the characteristics of the statistic. Estimator’s Characteristics

**8. **8 10.3 Estimating the Population Mean when the Population Variance is Known How is an interval estimator produced from a sampling distribution?
A sample of size n is drawn from the population, and its mean is calculated.
By the central limit theorem is normally distributed (or approximately normally distributed.), thus…

**9. **9 We have established before that 10.3 Estimating the Population Mean when the Population Variance is Known

**10. **10 This leads to the following equivalent statement The Confidence Interval for m ( s is known)

**11. **11 Interpreting the Confidence Interval for m

**12. **12 Graphical Demonstration of the Confidence Interval for m

**13. **13 The Confidence Interval for m ( s is known) Four commonly used confidence levels

**14. **14 The Confidence Interval for m ( s is known)

**15. **15 The Confidence Interval for m ( s is known)

**16. **16 The Confidence Interval for m ( s is known)

**17. **17 Example 10.1
Doll Computer Company delivers computers directly to its customers who order via the Internet.
To reduce inventory costs in its warehouses Doll employs an inventory model, that requires the estimate of the mean demand during lead time.
It is found that lead time demand is normally distributed with a standard deviation of 75 computers per lead time.
Estimate the lead time demand with 95% confidence. The Confidence Interval for m ( s is known)

**18. **18 Example 10.1 – Solution
The parameter to be estimated is m, the mean demand during lead time.
We need to compute the interval estimation for m.
From the data provided in file Xm10-01, the sample mean is
The Confidence Interval for m ( s is known)

**19. **19 Using Excel Tools > Data Analysis Plus > Z Estimate: Mean The Confidence Interval for m ( s is known)

**20. **20 Wide interval estimator provides little information. Information and the Width of the Interval

**21. **21 Wide interval estimator provides little information. Information and the Width of the Interval

**22. **22 The Width of the Confidence Interval

**23. **23 The Affects of s on the interval width

**24. **24 The Affects of Changing the Confidence Level

**25. **25 The Affects of Changing the Sample Size

**26. **26 10.4 Selecting the Sample size We can control the width of the confidence interval by changing the sample size.
Thus, we determine the interval width first, and derive the required sample size.
The phrase “estimate the mean to within W units”, translates to an interval estimate of the form

**27. **27 The required sample size to estimate the mean is
Click to see how the formula is developed. 10.4 Selecting the Sample size

**28. **28 Example 10.2
To estimate the amount of lumber that can be harvested in a tract of land, the mean diameter of trees in the tract must be estimated to within one inch with 99% confidence.
What sample size should be taken? Assume that diameters are normally distributed with s = 6 inches. Selecting the Sample size

**29. **29 Solution
The estimate accuracy is +/-1 inch. That is w = 1.
The confidence level 99% leads to a = .01, thus za/2 = z.005 = 2.575.
We compute Selecting the Sample size