6-1 Copyright © 2014, 2011, and 2008 Pearson Education, Inc.

6-2 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Statistics for Business and Economics Chapter 6 Inferences Based on a Single Sample

6-3 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Content 1.Identifying and Estimating the Target Parameter 2.Confidence Interval for a Population Mean: Normal (z) Statistic 3.Confidence Interval for a Population Mean: Student’s t-Statistic 4.Large-Sample Confidence Interval for a Population Proportion 5.Determining the Sample Size 6.Finite Population Correction for Simple Random Sampling 7.Confidence Interval for a Population Variance

6-4 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Learning Objectives 1.Estimate a population parameter (means, proportion, or variance) based on a large sample selected from the population 2.Use the sampling distribution of a statistic to form a confidence interval for the population parameter 3.Show how to select the proper sample size for estimating a population parameter

6-5 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Thinking Challenge Suppose you’re interested in the average amount of money that students in this class (the population) have on them. How would you find out?

6-6 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Statistical Methods Estimation Hypothesis Testing Inferential Statistics Descriptive Statistics

6-7 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. 6.1 Identifying and Estimating the Target Parameter

6-8 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Estimation Methods Estimation Interval Estimation Point Estimation

6-9 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Target Parameter The unknown population parameter (e.g., mean or proportion) that we are interested in estimating is called the target parameter.

6-10 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Target Parameter Determining the Target Parameter ParameterKey Words of PhraseType of Data µMean; averageQuantitative pProportion; percentage fraction; rateQualitative

6-11 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Point Estimator A point estimator of a population parameter is a rule or formula that tells us how to use the sample data to calculate a single number that can be used as an estimate of the target parameter.

6-12 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Point Estimation 1.Provides a single value Based on observations from one sample 2.Gives no information about how close the value is to the unknown population parameter 3.Example: Sample mean x = 3 is the point estimate of the unknown population mean

6-13 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Interval Estimator An interval estimator (or confidence interval) is a formula that tells us how to use the sample data to calculate an interval that estimates the target parameter.

6-14 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Interval Estimation 1.Provides a range of values Based on observations from one sample 2.Gives information about closeness to unknown population parameter Stated in terms of probability – Knowing exact closeness requires knowing unknown population parameter 3.Example: Unknown population mean lies between 50 and 70 with 95% confidence

6-15 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. 6.2 Confidence Interval for a Population Mean: Normal (z) Statistic

6-16 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Estimation Process Mean, , is unknown Population Sample Random Sample I am 95% confident that  is between 40 & 60. Mean x = 50

6-17 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Key Elements of Interval Estimation Sample statistic (point estimate) Confidence interval Confidence limit (lower) Confidence limit (upper) A confidence interval provides a range of plausible values for the population parameter.

6-18 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Confidence Interval According to the Central Limit Theorem, the sampling distribution of the sample mean is approximately normal for large samples. Let us calculate the interval estimator: That is, we form an interval from 1.96 standard deviations below the sample mean to 1.96 standard deviations above the mean. Prior to drawing the sample, what are the chances that this interval will enclose µ, the population mean?

6-19 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Confidence Interval If sample measurements yield a value of that falls between the two lines on either side of µ, then the interval will contain µ. The area under the normal curve between these two boundaries is exactly.95. Thus, the probability that a randomly selected interval will contain µ is equal to.95.

6-20 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Confidence Coefficient The confidence coefficient is the probability that a randomly selected confidence interval encloses the population parameter - that is, the relative frequency with which similarly constructed intervals enclose the population parameter when the estimator is used repeatedly a very large number of times. The confidence level is the confidence coefficient expressed as a percentage.

6-21 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. 95% Confidence Level If our confidence level is 95%, then in the long run, 95% of our confidence intervals will contain µ and 5% will not. For a confidence coefficient of 95%, the area in the two tails is.05. To choose a different confidence coefficient we increase or decrease the area (call it  ) assigned to the tails. If we place  /2 in each tail and z  /2 is the z-value, the confidence interval with coefficient (1 –  ) is

6-22 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Conditions Required for a Valid Large-Sample Confidence Interval for µ 1.A random sample is selected from the target population. 2.The sample size n is large (i.e., n ≥ 30). Due to the Central Limit Theorem, this condition guarantees that the sampling distribution of is approximately normal. Also, for large n, s will be a good estimator of .

6-23 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Large-Sample (1 –  )% Confidence Interval for µ where z  /2 is the z-value with an area  /2 to its right and in the standard normal distribution. The parameter  is the standard deviation of the sampled population, and n is the sample size. Note: When  is unknown and n is large (n ≥ 30), the confidence interval is approximately equal to where s is the sample standard deviation.

6-24 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Thinking Challenge You’re a Q/C inspector for Gallo. 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 © T/Maker Co. 2 liter

6-25 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Confidence Interval Solution*

6-26 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. 6.3 Confidence Interval for a Population Mean: Student’s t-Statistic

6-27 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Small Sample  Unknown Instead of using the standard normal statistic use the t–statistic in which the sample standard deviation, s, replaces the population standard deviation, .

6-28 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Student’s t-Statistic The t-statistic has a sampling distribution very much like that of the z-statistic: mound-shaped, symmetric, with mean 0. The primary difference between the sampling distributions of t and z is that the t-statistic is more variable than the z-statistic.

6-29 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Degrees of Freedom The actual amount of variability in the sampling distribution of t depends on the sample size n. A convenient way of expressing this dependence is to say that the t-statistic has (n – 1) degrees of freedom (df).

6-30 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. z t Student’s t Distribution 0 t (df = 5) Standard Normal t (df = 13) Bell-Shaped Symmetric ‘Fatter’ Tails

6-31 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. t - Table

6-32 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. t-value If we want the t-value with an area of.025 to its right and 4 df, we look in the table under the column t.025 for the entry in the row corresponding to 4 df. This entry is t.025 = The corresponding standard normal z-score is z.025 = 1.96.

6-33 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Small-Sample Confidence Interval for µ where t a/2 is based on (n – 1) degrees of freedom.

6-34 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Conditions Required for a Valid Small-Sample Confidence Interval for µ 1.A random sample is selected from the target population. 2.The population has a relative frequency distribution that is approximately normal.

6-35 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Estimation Example Mean (  Unknown) A random sample of n = 25 has = 50 and s = 8. Set up a 95% confidence interval estimate for .

6-36 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Thinking Challenge You’re a time study analyst in manufacturing. You’ve recorded the following task times (min.): 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?

6-37 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Confidence Interval Solution* x = 3.7 s = n = 6, df = n – 1 = 6 – 1 = 5 t.05 = 2.015

6-38 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. 6.4 Large-Sample Confidence Interval for a Population Proportion

6-39 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Sampling Distribution of 1.The mean of the sampling distribution of is p; that is, is an unbiased estimator of p. 3.For large samples, the sampling distribution of is approximately normal. A sample size is considered large if both 2.The standard deviation of the sampling distribution of is ; that is, where q = 1–p.

6-40 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Large-Sample Confidence Interval for where Note: When n is large, can approximate the value of p in the formula for.

6-41 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. 2.The sample size n is large. (This condition will be satisfied if both. Note that and are simply the number of successes and number of failures, respectively, in the sample.). Conditions Required for a Valid Large-Sample Confidence Interval for p 1.A random sample is selected from the target population.

6-42 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Estimation Example Proportion A random sample of 400 graduates showed 32 went to graduate school. Set up a 95% confidence interval estimate for p.

6-43 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Thinking Challenge You’re a production manager for a newspaper. You want to find the % defective. Of 200 newspapers, 35 had defects. What is the 90% confidence interval estimate of the population proportion defective?

6-44 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Confidence Interval Solution*

6-45 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Adjusted (1 –  )100% Confidence Interval for a Population Proportion, p where is the adjusted sample proportion of observations with the characteristic of interest, x is the number of successes in the sample, and n is the sample size.

6-46 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. 6.5 Determining the Sample Size

6-47 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Sampling Error In general, we express the reliability associated with a confidence interval for the population mean µ by specifying the sampling error within which we want to estimate µ with 100(1 – )% confidence. The sampling error (denoted SE), then, is equal to the half-width of the confidence interval.

6-48 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Sample Size Determination for 100(1 –  ) % Confidence Interval for µ In order to estimate µ with a sampling error (SE) and with 100(1 –  )% confidence, the required sample size is found as follows: The solution for n is given by the equation

6-49 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Sample Size Example What sample size is needed to be 90% confident the mean is within  5? A pilot study suggested that the standard deviation is 45.

6-50 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Sample Size Determination for 100(1 –  ) % Confidence Interval for p In order to estimate p with a sampling error SE and with 100(1 –  )% confidence, the required sample size is found by solving the following equation for n: The solution for n can be written as follows: Note: Always round n up to the nearest integer value.

6-51 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Sample Size Example What sample size is needed to estimate p within.03 with 90% confidence?

6-52 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Thinking Challenge 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?

6-53 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Sample Size Solution*

6-54 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Key Ideas Population Parameters, Estimators, and Standard Errors ParameterEstimatorStandard Error of Estimator Estimated Std Error Mean, µ Proportion, p

6-55 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Key Ideas Population Parameters, Estimators, and Standard Errors Confidence Interval: An interval that encloses an unknown population parameter with a certain level of confidence (1 –  ) Confidence Coefficient: The probability (1 –  ) that a randomly selected confidence interval encloses the true value of the population parameter.

6-56 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Key Ideas Key Words for Identifying the Target Parameter µ – Mean, Average p – Proportion, Fraction, Percentage, Rate, Probability  2 - Variance

6-57 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Key Ideas Commonly Used z-Values for a Large- Sample Confidence Interval 90% CI:(1 –  ) =.10z.05 = % CI:(1 –  ) =.05z.025 = % CI:(1 –  ) =.02z.005 = % CI:(1 –  ) =.01z.005 = 2.575

6-58 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Key Ideas Determining the Sample Size n Estimating µ: Estimating p:

6-59 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Key Ideas Finite Population Correction Factor Required when n/N >.05

6-60 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Key Ideas Confidence Interval for Population Variance Uses chi-square (  2 ) distribution Need to know and df.

6-61 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Key Ideas Illustrating the Notion of “95% Confidence”

6-62 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Key Ideas Illustrating the Notion of “95% Confidence”