Chapter 7 Estimates, Confidence Intervals, and Sample Sizes Inferences Based on a Single Sample Estimation with Confidence Intervals

Learning Objectives State what is estimated Distinguish point and interval estimates Explain interval estimates Estimating a population mean: s is known. Estimating a population mean: s is not known. Estimating a Population Proportion Compute Sample Size

Statistical Methods Statistical Methods Descriptive Statistics Inferential Estimation Hypothesis Testing 5

I am 95% confident that  is between 75 & 83. Estimation Process Mean, , is unknown  Population Random Sample Mean X= 79 Sample  I am 95% confident that  is between 75 & 83. 7

Unknown Population Parameters Are Estimated X Mean  Proportion p ^ Variance  2 s 2 Estimate Population Parameter... with the Sample Statistic

Source: Bureau of Labor Statistics  National Unemployment Rates,  2008 - 2015  Source: Bureau of Labor Statistics    Jan.  Feb.  Mar.  April  May  June  July  Aug.  Sept.  Oct.  Nov.  Dec.  2015   5.7   5.5 5.5 5.4 5.3 5.1  2014   6.6   6.7   6.3   6.1   6.2   5.9    5.8    5.6  2013   7.9   7.7   7.5   7.3   7.2    7.2    7.0    6.7  2012   8.3   8.3   8.2   8.1   8.2   8.1   7.8    7.9    7.8    7.8  2011   9.0   8.9   8.8   9.1   9.2    9.0    8.6    8.5  2010   9.7   9.9   9.5   9.6    9.6    9.8    9.4  2009   7.6   8.5   9.4   9.8  10.2  10.0  2008   4.9   4.8   5.1   5.0   5.6   5.8    6.6    6.8    7.2

Point Estimation Point Estimation Interval 14

Point Estimation A point estimate of a parameter (, p, or s 2) is the value of a statistic ( , , or s 2) used to estimate the parameter. p ^ X It provides a single value. It is based on observations from 1 sample. It gives no information about how close the value is to the unknown population parameter. Example: Sample mean = 79 is a point estimate of the unknown population mean. X

Interval Estimation Point Estimation Interval 14

Interval Estimation The sample mean rarely equals the population mean. That is, sampling error is to be expected. Therefore, in addition to the point estimate for  , we need to provide some information that indicates the accuracy of the point estimate. We do so by giving a confidence-interval estimate for .

Interval Estimation A confidence interval or interval estimate is a range of values used to estimate the true value of a population parameter. The interval is obtained from a point estimate of the parameter and a percentage that specifies the probability that the interval actually does contain the population parameter. This percentage is called the confidence level of the interval.

Confidence Level (CL) Is the probability (when given in decimal form) that the confidence interval contains the unknown population parameter. The CL is denoted (1 - ) That is, is the probability that the parameter is Not within the confidence interval. Typical values of CL are 99%, 95%, 90%. This means that the corresponding values of  are  = 0.01, 0.05, 0.10.

Key Elements of the CI The center of the interval is the point estimate E is called the margin of error and is half the length of the confidence interval. E is determined by a , s , and the sample size n. X .

Key Elements of the CI This interval contains the parameter m , (1-a)% of the times.

Margin of Error and CI

Interval Estimation Provides Range of Values. Is based on observations from 1 sample. Gives information about how close the estimate is the to unknown population parameter. Is stated in terms of probability. Example: unknown population mean lies between 66 and 70 with 95% confidence.

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Margin of Error or Interval Width

Factors Affecting Interval Width 1. As data dispersion measured by  increases the error or width E increases. As Sample Size n increases the error or width E decreases. 3. As the level of confidence (1 - ) % increases the width increases because it affects

Confidence Interval Estimates Intervals Mean Variance Proportion  Known  Unknown 43

CI for the Mean ( known) 1. Assumptions are Population standard deviation is known Population is normally distributed or Sampling distribution can be approximated by normal distribution (n  30) 2. Confidence Interval Estimate

Example 1 The mean of a random sample of n = 25 isX = 50. Set up a 95% confidence interval estimate for  if  = 10. 49

Example 2 You are a Quality Control inspector for Norton. The  for 2-liter bottles is .05 liters. A random sample of 100 bottles showedX = 1.99 liters. What is the 90% confidence interval estimate of the true mean amount in 2-liter bottles? 2 liter 2 litros Tinto

Solution 52

Confidence Interval Estimates Intervals Mean Variance Proportion  Known  Unknown 43

CI for the Mean ( unknown) If X is a normally distributed variable with mean μ and standard deviation σ, then, for samples of size n, the variableX is also normally distributed and has mean μ and standard deviation . Equivalently, the standardized version ofX , has the standard normal distribution.

CI for the Mean ( unknown) In practice, σ is unknown therefore we cannot base our CI procedure on the standardized version ofX. The best we can do is estimate σ using the sample standard deviation s and replace σ with s in the equation and base our CI procedure on the new variable,

t-Distributions and t-Curves The t-distribution depends on n and for each n there is t-curve. Notice, that to find a t-value you need to compute the sample mean and the sample standard deviation. This will usually require the use of a calculator.

Finding the t-Value Having a Specified Area to Its Right For a t-curve with 13 degrees of freedom, find t0.05; that is, find the t-value having area 0.05 to its right, as shown in the figure.

Finding the t-Value Having a Specified Area to Its Right To find the t-value in question, we use Table IV. For ease of reference, we have repeated a portion of Table IV in the next slide. Notice that the table provides the t-score only when you know the area to its right. Unlike the table for z-scores, the t-table cannot be used to find probabilities when you know what the t-score is. For this, you need to use the t-distribution in the calculator.

Finding the t-Value Having a Specified Area to Its Right For a t-curve with 13 degrees of freedom, find t0.05; that is, find the t-value having area 0.05 to its right, as shown in the figure.

Example 1 A random sample of n = 25 has X = 50 and s = 8. Set up a 95% confidence interval estimate for . 69

Example 2 You are a time study analyst in manufacturing. You have recorded the following task times (in minutes): 3.6, 4.2, 4.0, 3.5, 3.8, 3.1. What is the 90% confidence interval estimate of the population mean task time?

Solution 72

Finding Sample Sizes 9 88

Estimating the Sample Size

Example 1 What sample size is needed to be 90% confident of being correct within  5? A pilot study suggested that the standard deviation is 45. 91

Example 2 You work in Human Resources at Merrill Lynch. You plan to survey employees to find their average medical expenses. You want to be 95% confident that the sample mean is within ± $50. A pilot study showed that  was about $400. What sample size do you use?

Solution 93

Confidence Interval Estimates Intervals Mean Variance Proportion  Known  Unknown 43

Confidence Intervals for One Population Proportion 9 88

Proportion Notation and Terminology Many statistical studies are concerned with obtaining the proportion (percentage) of a population that has a specified attribute. For example, we might be interested in the percentage of U.S. adults who have health insurance, the percentage of cars in the US that are imports, the percentage of U.S. adults who favor stricter clean air health standards, or the percentage of Canadian women in the labor force.

Proportion Notation and Terminology Notice that in the previous examples, a given individual in the population will have the specified attribute or not. This means that we are interested in an experiment that can have only two possible outcomes. For instance, A U.S. adult does have health insurance or does not. A car in the U.S. is either an import or is not. etcetera

Proportion Notation and Terminology We introduced some notation and terminology used when we make inferences about a population proportion.

Proportion Notation and Terminology Sometimes we refer to x (the number of members in the sample that have the specified attribute) as the number of successes and to n − x (the number of members in the sample that do not have the specified attribute) as the number of failures.

Proportion Notation and Terminology Notice that for a given sample of size n, the quotient x/n, is the mean number of successes in n trials. That is, is the mean of the variable X which is 1 when the member in the sample has the attribute and 0 when the member does not. p ^

The Sampling Distribution of the Sample Proportion To make inferences about a population proportion p we need to know the sampling distribution of the sample proportion, that is, the distribution of the variable . Because a proportion can always be regarded as a mean, we can use our knowledge of the sampling distribution of the sample mean to derive the sampling distribution of the sample proportion. p ^

The accuracy of the normal approximation depends on n and p The accuracy of the normal approximation depends on n and p. If p is close to 0.5, the approximation is quite accurate, even for moderate n. The farther p is from 0.5, the larger n must be for the approximation to be accurate.

As a rule of thumb, we use the normal approximation when np and n(1 − p) are both 5 or greater. In this section, when we say that n is large, we mean that np and n(1 − p) are both 5 or greater.

Since in practice we do not know the value of p we replace the conditions np  5 and n(1− p)  5 with the conditions and This is the same as: the number of successes x and the number of failures n-x are both 5 or greater. np  5 ^ n(1− p)  5 ^

CI for the Proportion p

Margin of Error or Interval Width

Estimating the Sample Size The margin of error E and CL (1-a)% of a CI are often specified in advance. We must then determine the sample size required to meet those specifications. If we solve for n in the formula for E, we obtain This formula cannot be used to obtain the required sample size because the sample proportion, , is not known prior to sampling. p ^

Estimating the Sample Size The way around this problem is to observe that the largest can be is 0.25 when = 0.5 p (1-p) ^ p ^

Estimating the Sample Size p (1-p) ^ Because the largest possible value of is 0.25 the most conservative approach for determining sample size is to use that value in equation The sample size obtained then will generally be larger than necessary and the margin of error less than required. Nonetheless, this approach guarantees that the specifications will be met or bettered.

Estimating the Sample Size

Example 1 A random sample of 400 graduates showed 32 went to grad school. Set up a 95% confidence interval estimate for proportion p of students that go to grad school. 91

Example 2 You are a production manager for a newspaper. You want to find the % defective newspapers. Of 200 newspapers, 35 had defects. What is the 90% confidence interval estimate of the population proportion of defective newspapers? 91

Solution 91

Example 3 91

Example 4 91

Example 5 91