1. ©The McGraw-Hill Companies, Inc. 2008McGraw-Hill/Irwin Estimation and Confidence Intervals Chapter 9

2. 2 GOALS Define a point estimate. Define level of confidence. Construct a confidence interval for the population mean when the population standard deviation is known. Construct a confidence interval for a population mean when the population standard deviation is unknown. Construct a confidence interval for a population proportion. Determine the sample size for attribute and variable sampling.

3. 3 Point and Interval Estimates A point estimate is the statistic, computed from sample information, which is used to estimate the population parameter. – Sample mean for population mean. A confidence interval estimate is a range of values constructed from sample data so that the population parameter is likely to occur within that range at a specified probability. – The specified probability is called the level of confidence.

4. 4 Confidence Interval for a Mean (σ Known) We can get 95% (or 99%) confidence interval for a population mean, based on the CLT. – Mean of the sampling distribution is normally distributed with the mean μ and the variance σ 2 /n. – 95% of the sample means selected from a population will be within 1.96 standard deviations from the population mean μ. Here, the standard deviation means that of the sample means, i.e. the standard error. – From this, we can construct the 95% confidence interval for population mean (μ ). X(bar) +(-) 1.96 σ/n 1/2

5. 5 Factors Affecting Confidence Interval Estimates The factors that determine the width of a confidence interval are: 1.The sample size, n. 2.The variability in the population, usually σ estimated by s. 3.The desired level of confidence.

6. 6 Confidence Interval for a Mean (σ Known) In general, we can construct confidence interval for any confidence level (0-100%), by finding the corresponding value z. – 1.96 for 95% confidence level. – 2.58 for 99% confidence level.

7. 7 Interval Estimates- Example and Interpretation (pp. 298-299) Random sample of 256 managers’ income. A 95% confidence interval implies that about 95% of the similarly constructed intervals will contain the parameter being estimated. That is, the true population mean will belong to such intervals in 95 times out of 100 samples.

8. 8 Confidence Interval for a Mean (σ unknown) In many (most) of the cases, standard deviation of the population ( σ ) is unknown. – We cannot construct confidence interval using previous method (based on z, or standard normal distribution). – In this case, we may use sample standard deviation (s) to estimate the population standard deviation ( σ). – Then, we can show the following statistic follows so- called “t distribution”, which is similar to z (standard normal) distribution.

9. 9 Characteristics of the t-distribution 1. It is, like the z distribution, a continuous distribution. 2. It is, like the z distribution, bell-shaped and symmetrical. 3. There is not one t distribution, but rather a family of t distributions. All t distributions have a mean of 0, but their standard deviations differ according to the sample size, n. 4. The t distribution is more spread out and flatter at the center than the standard normal distribution * As the sample size increases, the t distribution approaches the standard normal distribution,

10. 10 Comparing the z and t Distributions when n is small

11. 11 Confidence Interval Estimates for the Mean Use Z-distribution If the population standard deviation is known. Use t-distribution If the population standard deviation is unknown.

12. 12 When to Use the z or t Distribution for Confidence Interval Computation

13. 13 Confidence Interval for the Mean – Example using the t-distribution A tire company investigates the tread life of its tires. A sample of 10 tires driven 50,000 miles revealed a sample mean of 0.32 inch of tread remaining with a standard deviation of 0.09 inch. Construct a 95 percent confidence interval for the population mean. Would it be reasonable for the company to conclude that after 50,000 miles the population mean of tread remaining is 0.30 inches?

14. 14 Student’s t-distribution Table

15. 15 Degrees of freedom (of t distribution) “df” (degrees of freedom): the number of observation in the sample (n) minus the number of samples. – In above case, n-1 (=9). – Why called “degree of freedom”? When sample statistics are used, it is necessary to determine the number of values that are free to vary. (eg) The mean of four numbers (7, 4, 1, 8) is 5. Here, the “df” is 3, because the sum of mean deviations is 0. That is, 1 degree of freedom is lost in the process of getting the sample mean. So refer to the row of distribution table with df=9.

16. 16 Confidence Interval Estimates for the Mean – Using Excel

17. 17 A Confidence Interval for a Proportion Proportion: the fraction, ratio, or percent indicating the part of the sample or the population having a particular trait of interest. Sample proportion: p = X/n – The population proportion is identified by π. (i.e. the proportion of successes in a binomial distribution) – If some conditions are met, then the p is following a normal distribution with mean π and standard deviation (p(1-p)/n) 1/2 approximately.

18. 18 Using the Normal Distribution to Approximate the Binomial Distribution For a confidence interval for a proportion to be constructed, the following assumptions need to be met. 1. The binomial conditions (Chapter 6), is to be met. Briefly, these conditions are: a. The sample data is the result of counts. b. There are only two possible outcomes (success or failure). c. The probability of a success remains the same from one trial to the next. d. The trials are independent, meaning the outcome on one trial does not affect the outcome on another. 2. The values nπ and n(1-π) should both be greater than or equal to 5. This condition allows us to employ the standard normal distribution z, to construct a confidence interval for a proportion.

19. 19 Confidence Interval for a Population Proportion The confidence interval for a population proportion ( π) is estimated by:

20. 20 Confidence Interval for a Population Proportion - Example The trade union representing company A is considering to merge with another union B. According to the union A’s bylaws, at least three-fourths of the union membership must approve any merger. A random sample of 2,000 current members reveals 1,600 plan to vote for the merger proposal. What is the estimate of the population proportion? Develop a 95 percent confidence interval for the population proportion. Basing your decision on this sample information, can you conclude that sufficient proportion of members favor the merger? Why?

21. 21 Finite-Population Correction Factor Previous discussion dealt with populations of very large size (or infinite) A population that has a fixed upper bound is said to be finite. For a finite population, where the total number of objects is N and the size of the sample is n, the following adjustment is made to the standard errors of the sample means and the proportion: However, if n/N <.05, the finite-population correction factor may be ignored.

22. 22 Effects on FPC when n/N Changes Observe that FPC approaches 1 when n/N becomes smaller

23. 23 Confidence Interval Formulas for Estimating Means and Proportions with Finite Population Correction C.I. for the Mean (  ) C.I. for the Proportion (  ) C.I. for the Mean (  )

24. 24 CI For Mean with FPC - Example There are 250 families in a town. A random sample of 40 of these families revealed the mean annual church contribution was $450 and the standard deviation of this was $75. Develop a 90 percent confidence interval for the population mean of church contribution. Interpret the confidence interval. Given in Problem: N – 250 n – 40 s - $75 Since n/N = 40/250 = 0.16, the finite population correction factor must be used. The population standard deviation is not known therefore use the t- distribution. Use the formula below to compute the confidence interval:

25. 25 CI For Mean with FPC - Example

26. 26 Choosing an Appropriate Sample Size How many items should be included in a sample? There are 3 factors that determine the appropriate size of a sample, none of which has direct relationship to the population size. The level of confidence desired. The maximum allowable error. The variation in the population. – Measured by population standard deviation.

27. 27 Sample Size Determination for a Variable To find the sample size for a variable: – Use the formula for margin of error, E= z ( σ/n 1/2 )

28. 28 A researcher wants to estimate the mean amount of remuneration for members of city councils in large cities. The error in estimating the mean is to be less than $100 with a 95 percent level of confidence. The researcher found a report by the Department of Labor that estimated the standard deviation to be $1,000. What is the required sample size? Given in the problem: E, the maximum allowable error, is $100 The value of z for a 95 percent level of confidence is 1.96, The estimate of the standard deviation is $1,000. Sample Size Determination for a Variable-Example

29. 29 Sample Size for Proportions The formula for determining the sample size in the case of a proportion is based on – the desired confidence level, margin of error (in the proportion), and an estimate of the proportion.

30. 30 Example for proportion A study needs to estimate the proportion of cities that have private refuse collectors. The investigator wants the margin of error to be within.10 of the population proportion, the desired level of confidence is 90 percent, and no estimate is available for the population proportion. What is the required sample size?

31. 31 End of Chapter 9