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Section 6-1 β Confidence Intervals for the Mean (Large Samples) β’ Estimating Population Parameters VOCABULARY: Point Estimate β a single value estimate for a population parameter. The most unbiased point estimate of the population parameter is the sample mean. Interval Estimate An interval, or range of values, used to estimate a population parameter. Level of Confidence - Denoted as c, it is the probability that the interval estimate contains the population parameter. Margin of Error Sometimes also called the maximum error of estimate, or error tolerance. It is denoted as E, and is the greatest possible distance between the point estimate and the value of the parameter it is estimating. c-confidence interval -Is found by adding and subtracting E from the sample mean. The probability that the confidence interval contains µ is c. FORMULAS: Margin of Error = π¬ = ππ ππ = ππ Ο π To use the formula, it is assumed that the population standard deviation is known. This is rarely the case, but when π β₯ 30, the sample standard deviation s can be used in place of Ο. s The formula effectively becomes π¬ = ππ ππ = InvNorm of 1βπ 1 , or (1 β 2 2 c-confidence interval sample standard deviation: π π) (explained why on bottom of page 311) πβπ¬<π<π+π¬ π= (πβπ₯)π πβπ GUIDELINES: 1)Find the sample statistics π and π₯. π is the sample size. π₯ is the sample mean. 2)Specify Ο, if known. Otherwise, if π β₯ 30, find the sample standard deviation s and use it as a point estimate for Ο. 3)Find the critical value π§π that corresponds to the given level of confidence. The three most commonly used confidence levels are 90%, 95%, and 99%. The corresponding π§π values are: 90% -- 1.645 95% -- 1.96 99% -- 2.576 It would be beneficial to memorize these. You will be using them a lot. GUIDELINES: 4)Find the margin of error E. (π¬ = ππ s ) π 5)Find the left and right endpoints and form the confidence interval. πβπ¬<π<π+π¬ Now the good news!! The TI-84 can help with this, too. We are going to walk through Example 4 on page 314, using the data points from Example 1 on page 310. 1)Enter the 50 data points into L1 on your calculator (STAT Edit). 2)STAT Calc 1 to find the sample standard deviation (we can use this because we have 50 data points; π β₯ 30). s = 5.01 3)STAT TESTS 7 (Z-Interval) 4)Select Data, since you have the data entered into the calculator. 5)Enter 5.01 as the standard deviation, and .99 as the C-Level (level of confidence). 5)Select Calculate to get the interval. We can be 99% sure that the actual population mean is between 10.575 and 14.225. Notice that the calculator also tells us that the mean of the data we entered is 12.4, that the standard deviation of the data was 5.01 and that n was 50. If we need to know what E is, simply find the distance between the interval endpoints and divide by 2. (14.225 β 10.575)/2 = 1.825. Look at Example 5 on page 315. n = 20, π₯ = 22.9, Ο = 1.5, and c = 90% STAT TESTS 7, select Stats (since you are going to provide the stats instead of the actual data points). Enter 1.5, 22.9, 20, and .9 and then calculate. We can be 90% certain that the actual mean age of all students currently enrolled in college is between 22.3 and 23.5. As a general rule, we round our interval endpoints to the same number of decimal points as the mean that is given to us. We were given 22.9, which is one decimal, so we rounded our interval to one decimal place. CALCULATING MINIMUM SAMPLE SIZE How do you know how many experiments or trials are needed in order to achieve the desired level of confidence for a given margin of error? We take the formula for finding E and solve it for n. s π Ο πΈ = ππ becomes π = ( π )2 . π πΈ Remember, if you donβt know what Ο is, you can use s, as long as π β₯ 30. Example 6 on page 316c = 95%, π§π = 1.96, π β π β 5.01 (from Example 1), E = 1 (given). π=( ππ Ο 2 1.96(5.0) 2 9.8 ) =( ) = ( )2 = πΈ 1 1 9.82 = 96.04. CALCULATING MINIMUM SAMPLE SIZE If you want to be 95% certain that the true population mean lies within the interval created with an E of 1, you need AT LEAST 97 magazine advertisements in your sample. We round up, since 96 advertisements are not quite enough. ASSIGNMENTS Classwork: Pages 317-318; #2-34 Evens Homework: Pages 317-323; #35-40 All, #45-67 Odd