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Section 7.4: Estimation of a population mean m (s is not known) This section presents methods for estimating a population mean when the population standard deviation s is not known. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 1 Sample Mean _ The sample mean x is still the best point estimate of the population mean m. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 2 Construction of a confidence intervals for m (s is not known) With σ unknown, we use the Student t distribution instead of normal distribution. It involves a new feature: number of degrees of freedom Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 3 Definition The number of degrees of freedom for a collection of sample data is the number of sample values that can vary after certain restrictions have been imposed on all data values. The degree of freedom is often abbreviated df. degrees of freedom = n – 1 in this section. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 4 Margin of Error E for Estimate of m (With σ Not Known) Formula 7-6 E = t / s 2 n where t/2 has n – 1 degrees of freedom. t/2 = critical t value separating an area of /2 in the right tail of the t distribution Table A-3 lists values for tα/2 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 5 Confidence Interval for the Estimate of μ (With σ Not Known) x–E <µ<x +E where E = t/2 s n df = n – 1 t/2 found in Table A-3 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 6 Important Properties of the Student t Distribution 1. The Student t distribution is different for different sample sizes (see the following slide, for the cases n = 3 and n = 12). 2. The Student t distribution has the same general symmetric bell shape as the standard normal distribution but it reflects the greater variability (with wider distributions) than that the standard normal distribution does. 3. The Student t distribution has a mean of t = 0 (just as the standard normal distribution has a mean of z = 0). 4. The standard deviation of the Student t distribution varies with the sample size and is greater than 1 (unlike the standard normal distribution, which has a s = 1). 5. As the sample size n gets larger, the Student t distribution gets closer to the normal distribution. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 7 Student t Distributions for n = 3 and n = 12 Figure 7-5 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 8 Choosing the Appropriate Distribution Use the normal (z) distribution Use t distribution s known and normally distributed population or s known and n > 30 s not known and normally distributed population or s not known and n > 30 Methods of Chapter 7 do not apply Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Population is not normally distributed and n ≤ 30 9 Confidence Intervals by TI-83/84 • Press STAT and select TESTS • Scroll down to TInterval press ENTER • choose Data or Stats. For Stats: _ • Type in x: (sample mean) Sx: (sample st. deviation) • n: (number of trials) • C-Level: (confidence level) • Press on Calculate • Read the confidence interval (…..,..…) Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 10 Confidence Intervals by TI-83/84 • Press STAT and select TESTS • Scroll down to TInterval press ENTER • choose Data or Stats. For Data: • Type in List: L1 (or L2 or L3) • (specify the list containing your data) Freq: 1 (leave it) • C-Level: (confidence level) • Press on Calculate • Read the confidence interval (…..,..…) Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 11 Finding the Point Estimate and E from a Confidence Interval Point estimate of µ: x = (upper confidence limit) + (lower confidence limit) 2 Margin of Error: E = (upper confidence limit) – (lower confidence limit) 2 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 12 Section 7-5 Estimating a Population Variance This section covers the estimation 2 of a population variance s and standard deviation s. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 13 Estimator of s 2 The sample variance s2 is the best point estimate of the population variance s2. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 14 Estimator of s The sample standard deviation s is a commonly used point estimate of s . Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 15 Construction of confidence 2 intervals for s We use the chi-square distribution, denoted by Greek character 2 (pronounced chi-square). Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 16 Properties of the Chi-Square Distribution 1. The chi-square distribution is not symmetric, unlike the normal and Student t distributions. degrees of freedom = n – 1 Chi-Square Distribution Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Chi-Square Distribution for df = 10 and df = 20 17 Properties of the Chi-Square Distribution 2. The values of chi-square can be zero or positive, but they cannot be negative. 3. The chi-square distribution is different for each number of degrees of freedom, which is df = n – 1. In Table A-4, each critical value of 2 corresponds to an area given in the top row of the table, and that area represents the cumulative area located to the right of the critical value. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 18 Example A sample of ten voltage levels is obtained. Construction of a confidence interval for the population standard deviation s requires the left and right critical values of 2 corresponding to a confidence level of 95% and a sample size of n = 10. Find the critical value of 2 separating an area of 0.025 in the left tail, and find the critical value of 2 separating an area of 0.025 in the right tail. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 19 Example Critical Values of the Chi-Square Distribution Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 20 Confidence Interval for Estimating a Population Variance (n 1)s 2 R 2 s Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 2 n 1)s ( 2 2 L 21 Confidence Interval for Estimating a Population Standard Deviation (n 1)s 2 R 2 s Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. (n 1)s 2 2 L 22 Requirement: The population must have normally distributed values (even if the sample is large) This requirement is very strict Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 23 Round-Off Rule for Confidence Intervals Used to Estimate s or s 2 1. When using the original set of data, round the confidence interval limits to one more decimal place than used in original set of data. 2. When the original set of data is unknown and only the summary statistics (n, x, s) are used, round the confidence interval limits to the same number of decimal places used for the sample standard deviation. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 24 Determining Sample Sizes The procedure for finding the sample size necessary to estimate s2 is based on Table 7-2. You just read the required sample size from an appropriate line of the table. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 25 Determining Sample Sizes Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 26 Example: We want to estimate the standard deviation s. We want to be 95% confident that our estimate is within 20% of the true value of s. How large should the sample be? Assume that the population is normally distributed. From Table 7-2, we can see that 95% confidence and an error of 20% for s correspond to a sample of size 48. We should obtain a sample of 48 values. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 27