Chapter 9 Estimation and Confidence Intervals

Our Objectives 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.

Point Estimate The statistic, computed from sample information, which is used to estimate the population parameter. Sample mean, X bar is a point estimate for the population mean, µ.

Confidence Interval 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. Ex. We are 90% sure that the mean yearly income of construction workers in the New York area is between $61,000 and $69,000.

Confidence Interval For reasonably large samples (n ≥ 30), the central limit theorem allows us to state the following: 95% of the sample means selected from a population will be within 1.96 SD’s of the population mean µ. 99% of the sample means will lie within 2.58 SD’s of the population mean.

Confidence Interval How did we get the 1.96 and the 2.58 for the 95% and 99% confidence intervals? For the 95% CI: 100 – 95 % = 5% remaining, divide it between 2 tails of SND curve for a =.4750 (or simply 95% / 2) In the table, the z value for.4750 is Use same reasoning and calculations for the 99% CI.

Confidence Interval Example: we get a sample (>30) of recent college graduates and compute the sample mean annual starting salary. It is $39,000. The SD (or the standard error) is $200. The 95% CI lies between what values? The confidence limits are: $39,000 ±1.96($200); $38,608 and $39,392

Standard Error Estimation When population SD is known: When population SD is unknown:

Confidence Interval for the Population Mean (n≥30)

Unknown Population SD & a Small Sample This situation is not covered by the central limit theorem. However, we can reason that the population is normal or reasonably close to a normal distribution. Under these conditions, we replace the standard normal distribution with the t distribution. The t distribution is a continuous distribution with many similarities to the standard normal distribution.

The Student’s t Distribution Developed in the early 1900’s by William Gosset, an English brewmaster. Gosset’s work was published under the pen name “Student”. Gosset was concerned with the behavior of the term: Gosset was worried about the discrepancy between s and σ when s was calculated from a very small sample.

Characteristics of 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 a family of t distributions. They all have a mean of 0. The SD differs according to sample size (SD larger with smaller n). 4. It is more spread out and flatter at the center than the standard normal distribution. As sample size increases, t distribution becomes closer to the standard normal distribution, because the errors in using s to estimate σ decrease with larger samples.

Confidence Interval for the Population Mean, σ unknown

When to Use t Distribution No Yes NoYes

Example A tire manufacturer would like to investigate the tread life of its tires. A sample of 10 tires driven 50,000 miles revealed a sample mean of.32 inch of tread remaining with a SD of.09 inch. Construct a 95% CI for the population mean. Would it be reasonable for the manufacturer to conclude that after 50,000 miles the population mean amount of tread remaining is.30 inches?

Example (cont’d) We assume the population distribution is normal. Since n=10, we use the formula To find the value of t, we use Appendix F. Locate the 95% CI column. Move down to df of 9 (10-1). The value in the cell is Substitute the values in the above formula: It is reasonable to conclude that the population mean is in this interval. The manufacturer can be 95% confident that the mean remaining tread depth is between and inches.

Proportion The fraction, ratio, or percent indicating the part of the sample or the population having a particular trait of interest. Used with nominal scale of measurement. Ex. A recent survey indicated that 92 out of 100 surveyed favored the continued use of daylight savings time in the summer. The sample proportion is 92/100, or.92, or 92%.

Assumptions for Proportion CI Construction 1. The binomial conditions have been met: a. Sample data is a result of counts. b. There are only 2 possible outcomes (Success and Failure). c. The probability of a success remains the same from one trial to the next. d. The trials are independent. 2. The values nπ and n(1-π) should be both ≥5. (π is the population proportion)

Sample Proportion Confidence Interval for a Population Proportion

Standard Error of the Sample Proportion Confidence Interval for a Population Proportion

Example The union representing ABC company is considering a merger with Teamsters Union. According to ABC union bylaws, at least three- fourth of the union membership must approve any merger. A random sample of 2,000 current ABC members reveal 1,600 plan to vote for the merger proposal. What is the estimate of the population proportion? Develop a 95% confidence interval for the population proportion. Basing your decision on this sample information, can you conclude that the necessary proportion of ABC members favor the merger? Why?

Example (cont’d) First calculate the sample proportion: P=X/N= 1,600/2,000=.80 80% of the population favor the merger proposal. We determine the 95% CI. The z value is The endpoints are.782 and.818. The lower limit is greater than.75. So, we conclude that the merger proposal will likely pass because the interval estimate includes values greater than 75% of the union membership.

Finite Population A population that has a fixed upper bound limit. A small population. Ex. Students registered for this class. When the sampled population is finite, we need to make some adjustments in the way we compute the standard error of the sample means, and the standard error of the sample proportion. The adjustment is called the finite-population correction factor.

Finite-Population Correction Factor Standard error of the sample mean: Standard error of the sample proportion: N is the total number of objects in the population, n is the sample size. p is the sample proportion. The usual rule for correction factor use is: If the ratio of n/N <.05, the correction factor is ignored.

Choosing an Appropriate Sample Size Sample size depends on three factors: 1. The level of confidence desired. 2. The margin of error the researcher will tolerate. 3. The variability in the population being studied.

1. Researchers set the level of confidence desired. The 95% and 99% levels of confidence are the most common, but any value between 0 and 100 percent is possible. The higher the level of confidence selected, the larger the size of the corresponding sample. Choosing an Appropriate Sample Size

2. The maximum allowable error Is the amount added and subtracted to the sample mean to determine the endpoints for the CI. It is the amount of error the researchers are willing to tolerate. A small allowable error will require a large sample. A large allowable error will permit a small sample size. Choosing an Appropriate Sample Size

3. The population SD If the population is widely dispersed, a large sample is required. If the population is concentrated (homogeneous), the required sample size will be smaller. It may be necessary to estimate the population SD. Choosing an Appropriate Sample Size

Estimate for Population SD 1. Use a comparable study. 2. Use a range-based approach. Estimate highest and smallest values in population. Distance between those values, the range, is 6 SD’s. Therefore, the SD is 1/6 th of that range. 3. Conduct a pilot study. This is the most common method.

Sample Size for Estimating Population Mean n is the sample size; z is the standard normal value corresponding to the desired level of confidence; s is an estimate of the population SD; E is the maximum allowable error. If the result is not a whole number, round up.

Example A student wants to determine the mean amount of earnings per month of city council members. The error in estimating the mean is to be less than $100 with a 95% level of confidence. The student found a report by the Department of Labor that estimated the SD to be $1,000. What is the required sample size?

Example (cont’d) The maximum allowable error, E, is $100. The value of z for a 95% level of confidence is 1.96, and the estimate for the SD is $1,000. substituting these values in the formula: A sample of 385 is required to meet the specifications.

Example (cont’d) What if the student wanted to increase the level of confidence to 99%? The corresponding z value is The recommended sample size is now 666. Notice the change in the required sample size for the different levels of confidence. There is an increase of 281 observations. This could greatly increase the cost and the time of the study. Therefore, the level of confidence should be considered carefully.

Sample Size for the Population Proportion Three items need to be specified: 1.The desired level of confidence. 2.The margin of error in the population proportion. 3.An estimate of the population proportion. If an estimate of π (or p) is available from a pilot study we can use it in the formula; if not, we use p=.5.

Example A group of students want to estimate the proportion of cities with subsidized transportation systems. They want the estimate to be within.10 of the population proportion. The desired level of confidence is 90%. No estimate for the population proportion is available. What is the required sample size?

Example (cont’d) E=.10 The level of confidence is 90%. The corresponding z value is No estimate for p is available, so we use.50. They need a random sample of 69 cities.

Homework 12 th edition: 31, 32, 33, 37 (pg. 306), 45 (pg. 307), 55 (pg. 308). 13 th edition: 31, 32, 33, 37 (pg.319), 45 (pg.320), 59 (pg.322).