7.1 - 1 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Chapter 7 Estimates and Sample Sizes 7-1 Review and Preview 7-2 Estimating.

7.1 - 1 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Chapter 7 Estimates and Sample Sizes 7-1 Review and Preview 7-2 Estimating a Population Proportion 7-3 Estimating a Population Mean: σ Known (OMIT) 7-4 Estimating a Population Mean: σ Not Known 7-5 Estimating a Population Variance (OMIT)

7.1 - 3 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Review  Chapters 2 & 3 we used “descriptive statistics” when we summarized data using tools such as graphs, and statistics such as the mean and standard deviation.  Chapter 6 we introduced critical values: z  denotes the z score with an area of  to its right. If  = 0.025, the critical value is z 0.025 = 1.96. That is, the critical value z 0.025 = 1.96 has an area of 0.025 to its right.

7.1 - 4 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Preview  The two major activities of inferential statistics are (1) to use sample data to estimate values of a population parameters, and (2) to test hypotheses or claims made about population parameters.  We introduce methods for estimating values of these important population parameters: proportions, means, and variances.  We also present methods for determining sample sizes necessary to estimate those parameters. This chapter presents the beginning of inferential statistics.

7.1 - 6 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Key Concept In this section we present methods for using a sample proportion to estimate the value of a population proportion. The sample proportion is the best point estimate of the population proportion. We can use a sample proportion to construct a confidence interval to estimate the true value of a population proportion, and we should know how to interpret such confidence intervals. We should know how to find the sample size necessary to estimate a population proportion.

7.1 - 9 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Example: Because the sample proportion is the best point estimate of the population proportion, we conclude that the best point estimate of p is 0.70. When using the sample results to estimate the percentage of all adults in the United States who believe in global warming, the best estimate is 70%. In the Chapter Problem (page 314) we noted that in a Pew Research Center poll, 70% of 1501 randomly selected adults in the United States believe in global warming, so the sample proportion is = 0.70. Find the best point estimate of the proportion of all adults in the United States who believe in global warming.

7.1 - 10 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Definition A confidence interval (or interval estimate) is a range (or an interval) of values used to estimate the true value of a population parameter. A confidence interval is sometimes abbreviated as CI. Here is an example of a confidence interval for the population proportion parameter:

7.1 - 12 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. We must be careful to interpret confidence intervals correctly. There is a correct interpretation and many different and creative incorrect interpretations of the confidence interval a < p < b Typically, we interpret the 95% confidence interval as follows: “We are 95% confident that the interval from a to b actually does contain the true value of the population proportion p.” Interpreting a Confidence Interval

7.1 - 13 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. This means that if we were to select many different samples of the same size and construct the corresponding confidence intervals, 95% of them would actually contain the value of the population proportion p. (Note that in this correct interpretation, the level of 95% refers to the success rate of the process being used to estimate the proportion.) Interpreting a Confidence Interval

7.1 - 14 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. For example, if we calculate the 95% confidence intervals for 20 different samples of a population, we expect that 95% of the 20 samples, or 19 samples, would have confidence intervals that contain the true value of p. Interpreting a Confidence Interval

7.1 - 15 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Consider the chapter problem example (global warming) again. Suppose we know the true proportion of all adults who believe in global warming is p=0.75 With 95% confidence interval, if we sample 20 times we may compute a confidence interval which does not actually contain p=0.75, such as, but, 19 times out of 20 we would find confidence intervals that do contain p=0.75. This is illustrated in Figure 7-1. Interpreting a Confidence Interval

7.1 - 19 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Critical Values A standard z score can be used to distinguish between sample statistics that are likely to occur and those that are unlikely to occur. Such a z score is called a critical value. Critical values are based on the following observations: Under certain conditions, the sampling distribution of sample proportions can be approximated by a normal distribution.

7.1 - 20 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Critical Values DEFINE: A z score of associated with a sample proportion has a probability of  /2 of falling in the right tail. Therefore:  To find, find the z-score in Table A-2 that corresponds to an area of

7.1 - 23 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Definition A critical value is the number on the borderline separating sample statistics that are likely to occur from those that are unlikely to occur. The number z  /2 is a critical value that is a z score with the property that it separates an area of  /2 in the right tail of the standard normal distribution.

7.1 - 24 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Critical Value Because the standard normal distribution is symmetric about the value of z=0, the value of – z  / 2 is at the vertical boundary for the area of  / 2 in the left tail

7.1 - 26 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. A confidence level is the probability 1 –  (often expressed as the equivalent percentage value) that the confidence interval actually does contain the population parameter, assuming that the estimation process is repeated a large number of times. (The confidence level is also called degree of confidence, or the confidence coefficient.) Definition

7.1 - 27 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Common choices for confidence levels are: 90% confidence level where  = 10%, 95% confidence level where  = 5%, 99% confidence level where  = 1%, Definition

7.1 - 28 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Confidence Level For the standard normal distribution a confidence level of P % corresponds to P percent of the area between the values and For example, if, the confidence level is 1-0.05=0.95=95% which gives the z-score and 95% of the area lies between -1.96 and 1.96 –z/2–z/2 z/2z/2  = 5%

7.1 - 31 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Definition When data from a simple random sample are used to estimate a population proportion p, the margin of error, denoted by E, is the maximum likely difference (with probability 1 – , such as 0.95) between the observed proportion and the true value of the population proportion p. The margin of error E is also called the maximum error of the estimate and can be found by multiplying the critical value and the standard deviation of the sample proportions:

7.1 - 33 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. p = population proportion Confidence Interval for Estimating a Population Proportion p = sample proportion n = number of sample values E = margin of error z  /2 = z score separating an area of  /2 in the right tail of the standard normal distribution

7.1 - 34 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Requirements for Using a Confidence Interval for Estimating a Population Proportion p 1. The sample is a simple random sample. 2. The conditions for the binomial distribution are satisfied: there is a fixed number of trials, the trials are independent, there are two categories of outcomes, and the probabilities remain constant for each trial. 3. There are at least 5 successes and 5 failures.

7.1 - 37 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Round-Off Rule for Confidence Interval Estimates of p Round the confidence interval limits for p to three significant digits.

7.1 - 42 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Calculator Use Here is what the solution manual suggests Calculate first To use the formula on a TI calculator:  Calculate, press the multiply key, the parentheses key, then 1- ANS (ANS is the 2 nd (-) key on bottom row), then ENTER. This will give z/2z/2

7.1 - 43 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Example  Press the divide key and input the value of n then ENTER. This will give  Press the square root key, then ANS, then ENTER. This will give  Finally press the multiply key then input the value of then ENTER. This will give E z/2z/2

7.1 - 44 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 1.Verify that the required assumptions are satisfied. (The sample is a simple random sample, the conditions for the binomial distribution are satisfied, and the normal distribution can be used to approximate the distribution of sample proportions because np  5, and nq  5 are both satisfied.) 2. Refer to Table A-2 and find the critical value z  /2 that corresponds to the desired confidence level. 3. Evaluate the margin of error Procedure for Constructing a Confidence Interval for p

7.1 - 45 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 4. Using the value of the calculated margin of error, E and the value of the sample proportion, p, find the values of p – E and p + E. Substitute those values in the general format for the confidence interval: ˆ ˆ ˆ p – E < p < p + E ˆ ˆ 5. Round the resulting confidence interval limits to three significant digits. Procedure for Constructing a Confidence Interval for p - cont

7.1 - 54 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Example Page 328, problem 10 ANSWER: The sample proportion is the midpoint of the upper and lower limits of the confidence interval

7.1 - 55 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Example Page 328, problem 10 ANSWER: The margin of error is the difference between the upper limit of the confidence interval and the sample proportion

7.1 - 57 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Calculate Confidence Intervals Directly From Calculator The TI calculator will compute confidence intervals as follows:  Press STAT and select TESTS  Select A:1-PropZInt  Enter x,n,C-Level and then calculate You should be able to calculate a confidence interval both ways: use the formula as in previous examples and directly with A:1- PropZInt as above

7.1 - 64 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Example Page 329, problem 30 c) We interpret the answer to part (b) as follows: “We are 99% confident that the true value of the proportion of boys in the population will be between 0.758 and 0.913.” Therefore,

7.1 - 65 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Example Page 329, problem 30 c) if the YSORT method has no effect we expect the population proportion to be p=0.5 which is not within the 99% confidence interval from part (b) and we can be 99% confident that the YSORT method is effective.

7.1 - 68 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Example Page 329, problem 34 b) We interpret the answer to part (a) as follows: “We are 99% confident that the true value of the proportion of people who say they vote in the population will be between 0.662 and 0.737.” Therefore,

7.1 - 69 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Example Page 329, problem 34 c) since the true population proportion is given as p=0.61 which is not within the 99% confidence interval from part (b), we can be 99% confident that people do not tell the truth about their voting record.

7.1 - 70 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Analyzing Polls When analyzing polls consider: 1.The sample should be a simple random sample, not an inappropriate sample (such as a voluntary response sample). 2.The confidence level should be provided. (It is often 95%, but media reports often neglect to identify it.) 3.The sample size should be provided. (It is usually provided by the media, but not always.) 4.Except for relatively rare cases, the quality of the poll results depends on the sampling method and the size of the sample, but the size of the population is usually not a factor.

7.1 - 71 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Caution Never follow the common misconception that poll results are unreliable if the sample size is a small percentage of the population size. The population size is usually not a factor in determining the reliability of a poll.

7.1 - 72 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Sample Size Suppose we want to collect sample data in order to estimate some population proportion. The question is how many sample items must be obtained?

7.1 - 75 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Sample Size for Estimating Proportion p When no estimate of p is known: NOTE: here we are assuming that ( ) 2 0.25 n = E 2  z ˆ

7.1 - 76 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Round-Off Rule for Determining Sample Size If the computed sample size n is not a whole number, round the value of n up to the next larger whole number.

7.1 - 78 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Example Page 331, problem 42 a)We are told that the sample percentage is within 4 percentage points of the true population percentage. This means that the margin of error is E=0.04

7.1 - 81 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Example Page 331, problem 42 b) sample proportion is given and the same values for the critical value and margin of error as in part (a) to get

7.1 - 82 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Recap In this section we have discussed:  Point estimates.  Confidence intervals.  Confidence levels.  Critical values.  Margin of error.  Determining sample sizes.

7.1 - 84 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Key Concept This section presents methods for estimating a population mean when the population standard deviation is not known. With σ unknown, we use the Student t distribution assuming that the relevant requirements are satisfied.

7.1 - 87 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. If the distribution of a population is essentially normal, then the distribution of is a Student t Distribution for all samples of size n. It is often referred to as a t distribution and is used to find critical values denoted by t  /2. t = x - µ s n Student t Distribution

7.1 - 88 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 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) that is expected with small samples. 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  = 1). 5.As the sample size n gets larger, the Student t distribution gets closer to the normal distribution.

7.1 - 90 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. degrees of freedom = n – 1 in this section. 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.

7.1 - 91 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. E = margin of error t  /2 = critical t value separating an area of  /2 in the right tail of the t distribution Notation

7.1 - 92 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Margin of Error E for Estimate of  (With  σ Not Known) Formula 7-6 where t  2 has n – 1 degrees of freedom. n s E = t   2 Table A-3 lists values for t α/2

7.1 - 94 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. where E = t  /2 n s x – E < µ < x + E t  /2 found in Table A-3 Confidence Interval for the Estimate of μ (With σ Not Known) df = n – 1

7.1 - 95 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Example Use the sample statistics of n = 49, = 0.4 and s = 21.0 to construct a 95% confidence interval estimate of the population mean.

7.1 - 96 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. With n = 49, the df = 49 – 1 = 48 Closest df in Table A-3 is 50, using two tails  = 5%=0.05 using one tail  /2= 2.5%=0.025 t  /2 = 2.009 Example 95% confidence level so  = 5%=0.05

7.1 - 98 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Requirements for Using a Confidence Interval for Estimating a Population Mean µ 1. The sample is a simple random sample. 2. Either the sample is from a normally distributed population or n>30

7.1 - 99 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 2. Using n – 1 degrees of freedom, refer to Table A-3 or use technology to find the critical value t  2 that corresponds to the desired confidence level. Procedure for Constructing a Confidence Interval for µ (With σ Unknown) 1. Verify that the requirements are satisfied. 3. Evaluate the margin of error E = t  2 s / n. 4. Find the values of Substitute those values in the general format for the confidence interval: 5. Round the resulting confidence interval limits.

7.1 - 100 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Round-Off Rule for Confidence Interval Estimates of µ  If using the original set of data, round to one more decimal place than is used for the original data set.  If using summary statistics,Sx, n round to the same number of decimal places used for the sample mean

7.1 - 101 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. You will not be given Table A-3 on the exam and should be able to use your calculator to compute the confidence interval for population mean with σ unknown. This is the method that will be used for the remaining slides. CALCULATOR

7.1 - 103 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Example The TI calculator will compute confidence intervals for population mean with σ unknown as follows:  Press STAT and select TESTS  Select 8:Tinterval  Arrow right to select Inpt: Stats  Enter,Sx, n, C-Level and then calculate

7.1 - 104 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Example Page 354, problem 14 ANSWER to part (b)  Press STAT and select TESTS  Select 8:Tinterval  Arrow right to select Inpt: Stats  Enter  Calculate then gives: (0.06395,0.17605)

7.1 - 105 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Example Page 354, problem 14 ANSWER to part (b)  We must round two decimal places (same number of places as the sample mean)  Units are grams/mile

7.1 - 106 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Point estimate of µ : x = (upper confidence limit) + (lower confidence limit) 2 Margin of Error: E = upper confidence limit - x Finding the Point Estimate and E from a Confidence Interval

7.1 - 110 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Example Page 354, problem 18b)  Press STAT and select TESTS  Select 8:Tinterval  Arrow right to select Inpt: Stats  Enter then calculate to get the confidence interval (3002.3,3203.7) which rounds to:

7.1 - 111 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Example Page 354, problem 18 c) ANSWER: Yes, since the confidence interval for the mean birth weight for mothers who used cocaine is entirely below the confidence interval in part (b) for mothers who did not use cocaine, it appears that cocaine use is associated with lower birth weights.

7.1 - 112 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Confidence Intervals for Comparing Data As in Sections 7-2 and 7-3, confidence intervals can be used informally to compare different data sets, but the overlapping of confidence intervals should not be used for making formal and final conclusions about equality of means.

7.1 - 114 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Example Page 355, problem 22(a)  Press STAT and select TESTS  Select 8:Tinterval  Arrow right to select Inpt: Stats  Enter then calculate to get the confidence interval

7.1 - 115 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Example Page 355, problem 22(b)  Press STAT and select TESTS  Select 8:Tinterval  Arrow right to select Inpt: Stats  Enter then calculate to get the confidence interval

7.1 - 118 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Example Page 356, problem 26  Here: and we are not given the mean and standard deviation. We must determine these from the given data.

7.1 - 121 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Example 0.06-0.0610.003721 0.11-0.0110.000121 0.160.0390.001521 0.150.0290.000841 0.140.0190.000361 0.08-0.0410.001681 0.150.0290.000841 Sum of last column:

3.1 - 122 Calculator: 1)Enter the list of data values into a list using STAT 1:Edit 2)Select 2nd STAT (LIST) and arrow right to choose MATH option 3:mean( 3)Select 2nd STAT (LIST) and arrow right to choose MATH option 7:stdDev( 4)Choose the list the data is in

7.1 - 123 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Example  Press STAT and select TESTS  Select 8:Tinterval  Arrow right to select Inpt: Stats  Enter then calculate to get the confidence interval

7.1 - 125 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Recap In this section we have discussed:  Student t distribution.  Degrees of freedom.  Margin of error.  Confidence intervals for μ with σ unknown.  Choosing the appropriate distribution.  Point estimates.  Using confidence intervals to compare data.

7.1 - 1 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Chapter 7 Estimates and Sample Sizes 7-1 Review and Preview 7-2 Estimating.

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7.1 - 1 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Chapter 7 Estimates and Sample Sizes 7-1 Review and Preview 7-2 Estimating.

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Presentation on theme: "7.1 - 1 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Chapter 7 Estimates and Sample Sizes 7-1 Review and Preview 7-2 Estimating."— Presentation transcript:

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