Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall 8 Chapter Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall

Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall Confidence Intervals Confidence Intervals Population Mean Population Proportion Section 8.4 σ Known σ Unknown Section 8.2 Section 8.3 Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall

Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall 8.2 Calculating Confidence Intervals for the Mean when the Standard Deviation (σ) of a Population is Known A confidence interval for the mean interval (or range) around the sample mean ( ) within which true (population) mean (µ) is expected to be µ = 362.3 ± 1.96 * 15 = A confidence level (dependent on Z(α)) probability that the interval estimate will include the population parameter of interest (1-α) Here α = .05 => 95% confidence Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall

Confidence Intervals for the Mean, σ Known Assumptions for section 8.2: The sample size is at least 30 (n ≥ 30) The population standard deviation (σ) is known Recall from Chapter 7: Formula for the Standard Error of the Mean where σ = Population standard deviation n = Sample size Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall

Confidence Intervals for the Mean, σ Known Formulas for the Confidence Interval for the Mean (σ Known) Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall

Confidence Intervals for the Mean, σ Known is called the critical z-score The variable is known as the significance level Example: if = .10, then = 1.645 is the value that encloses 90% of the area under the normal distribution and leaves 5% in each tail The total area to the left of the right-hand boundary is 0.90 + 0.05 = 0.95 The total area to the left of the left-hand boundary is 0.05 α/2 = 0.05 α/2 = 0.05 0.90 -1.645 1.645 z 0.95 0.05 Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall

Calculating the Margin of Error The Margin of Error ME is the amount added and subtracted to the point estimate to form the confidence interval Margin of Error Margin of Error Lower Confidence Limit Upper Confidence Limit Point Estimate Width of confidence interval Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall

Calculating the Margin of Error Increasing the sample size while keeping the confidence level constant will reduce the margin of error, resulting in a narrower (more precise) confidence interval Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall

Interpreting a Confidence Interval We are 90% confident that the true mean is between 137.10 and 153.90 Population mean (µ) may/may not be in this interval 90% of the sample means drawn from samples of this population will produce confidence intervals that include that population’s mean An incorrect interpretation is that there is 90% probability that this interval contains the true population mean (This interval either does or does not contain the true mean, there is no probability for a single interval) Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall

Interpreting a Confidence Interval Each interval extends from to But varies from sample to sample x1 x2 For 90% confidence, 90% of intervals constructed contain μ ; 10% do not Confidence Intervals Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall

Changing Confidence Levels The significance level, , Probability any given confidence interval will not contain the true population mean The confidence level of an interval is the complement to the significance level, 1 ─ i.e., a 100(1 – )% confidence interval has a significance level equal to The confidence interval gets wider if the confidence level increases (as Z(α) increases) Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall

Changing Confidence Levels Consider a 95% confidence interval: 0.975 encloses 95% of the area under the curve, with 2.5% in each tail Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall

Changing Confidence Levels z-scores for the most commonly used confidence levels are shown in this table: Confidence Level: Significance Level: Critical z-score: 80% 90% 95% 98% 99% 20% 10% 5% 2% 1% z0.10 = ±1.28 z0.05 = ±1.645 z0.025 = ±1.96 z0.01 = ±2.33 z0.005 = ±2.575 Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall

Using Excel to Determine Confidence Intervals for the Mean (σ Known) Excel’s CONFIDENCE function calculates the margin of error for confidence intervals The CONFIDENCE function has the following characteristics: =CONFIDENCE (alpha, standard_dev, size) where: alpha = The significance level of the confidence interval standard_dev = The standard deviation of the population size = Sample size Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall

Confidence Intervals for the Mean with Small Samples when σ is Known When the sample size is less than 30 and sigma is known, the population must be normally distributed to calculate a confidence interval With n < 30 the Central Limit Theorem cannot be applied, so we can’t say the sampling distribution will be approximately normal… …but the sampling distribution is always normal (regardless of sample size) if the population is normally distributed Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall

Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall 8.3 Calculating Confidence Intervals when the Population Standard Deviation (σ) is Unknown Population standard deviation is unknown, Use s, the sample standard deviation, in its place calculate the standard error (of the mean) Formula for the Sample Standard Deviation (recall from Chapter 3): where: x = The sample mean n = The sample size (number of data values) (xi – x ) = The difference between each data value and the sample mean Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall

Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall Confidence Intervals Confidence Intervals Population Mean Population Proportion Section 8.4 σ Known σ Unknown Section 8.2 Section 8.3 Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall

Using the Student’s t-distribution Formula for the Approximate Standard Error of the Mean The Student’s t-distribution used instead of normal probability distribution when the sample standard deviation, s, is used in place of the population standard deviation, σ Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall

Using the Student’s t-distribution Formulas for the Confidence Interval for the Mean (σ Unknown) where: = The sample mean = The critical t-score = The approximate standard error of the mean Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall

Using the Student’s t-distribution Normal distribution t (df = 13) t-distributions are bell-shaped and symmetric, but have ‘fatter’ tails than the normal t (df = 5) t Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall

Using the Student’s t-distribution The t-distribution is a continuous probability distribution with the following properties: It is bell-shaped and symmetrical around the mean The shape of the curve depends on the degrees of freedom (df), df = n – 1 The area under the curve is equal to 1.0 The t-distribution is flatter and wider than the normal distribution The critical score for the t-distribution is greater than the critical z-score for the same confidence level Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall

Using the Student’s t-distribution Comparing t-scores and z-scores: Confidence t t t z Level (10 df ) (20 df ) (30 df ) .80 1.372 1.325 1.310 1.28 .90 1.812 1.725 1.697 1.645 .95 2.228 2.086 2.042 1.96 .99 3.169 2.845 2.750 2.575 Note: t z as n increases Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall

Using the Student’s t-distribution The t-distribution is actually a family of distributions. As the number of degrees of freedom increases, the shape of the t-distribution becomes similar to the normal distribution With more than 100 degrees of freedom (a sample size of more than 100), the two distributions are practically identical Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall

Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall Using Excel and PHStat2 to Determine Confidence Intervals for the Mean (σ Unknown) The critical t-score can be found with Excel’s TINV function, which has the following characteristics: =TINV (alpha, degrees of freedom) where: alpha ( ) = The significance level of the confidence interval degrees of freedom = n - 1 n = Sample size Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall

Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall Confidence Intervals Confidence Intervals Population Mean Population Proportion Section 8.4 σ Known σ Unknown Section 8.2 Section 8.3 Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall

8.4 Calculating Confidence Intervals for Proportions Proportion data follow the binomial distribution, which can be approximated by the normal distribution under the following conditions: nπ ≥ 5 and n(1 – π) ≥ 5 where: π = The probability of a success in the population n = The sample size Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall

Calculating Confidence Intervals for Proportions The confidence interval for the proportion is an interval estimate around a sample proportion that provides us with a range of where the true population proportion lies Formula for the Sample Proportion: Formula for the Standard Error of the Proportion: where: π = The population proportion x = The number of observations of interest in the sample (successes) n = The sample size Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall

Calculating Confidence Intervals for Proportions Since the population proportion π is unknown, it is estimated using the sample proportion, p Formula for the Approximate Standard Error of the Proportion: where: p = The sample proportion n = The sample size Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall

Calculating Confidence Intervals for Proportions Formulas for the Confidence Interval for a Proportion: where: p = The sample proportion = The critical z-score = The approximate standard error of the proportion Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall

Calculating Confidence Intervals for Proportions Formula for the Margin of Error for a Confidence Interval for the Proportion Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall

Calculating Confidence Intervals for Proportions Example: From a random sample of U.S. citizens, 22 of 100 people are found to have blue eyes. Calculate a 98% confidence interval for the population proportion of blue eyes for U.S. citizens Calculate the sample proportion and the approximate standard error of the proportion: Find for 98%: 3. Calculate the interval endpoints: (next slide) Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall

8.5 Determining the Sample Size Increasing the sample size, holding all else constant, reduces the margin of error and provides a narrower confidence interval The sample size needed to achieve a specific margin of error can be calculated, given the following information: The confidence level The population standard deviation Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall

Calculating the Sample Size to Estimate a Population Mean Solving for n: Formula for the Sample Size Needed to Estimate a Population Mean: Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall

Calculating the Sample Size Needed to Estimate a Population Proportion Solving for n: Formula for the Sample Size Needed to Estimate a Population Proportion: Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall

Calculating the Sample Size Needed to Estimate a Population Proportion In order to calculate the required sample size to estimate π, the population proportion, we need to know p, the sample proportion Select a pilot sample and use the sample proportion, p If it is not possible to select a pilot sample, set p = 0.5 Setting it equal to 0.50 provides the most conservative estimate for a sample size (a sample size that is at least large enough to satisfy the margin-of-error requirement) Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall

Calculating the Sample Size Needed to Estimate a Population Proportion Example: What sample size is needed to estimate with 95% confidence the population proportion of U.S. citizens with blue eyes within a margin of error of ± 5%? Assume a pilot sample of 100 people found 22 with blue eyes. Find for 95%: Calculate the required sample size: so use a sample of size n = 264 (always round up) Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall

Calculating Confidence Intervals for Finite Populations Formulas for the Confidence Interval for a Proportion of a Finite Population Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall