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Chapter 7 Estimates and Sample Sizes

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). Section 7-1 Review and Preview. Review.

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Chapter 7 Estimates and Sample Sizes

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  1. Chapter 7Estimates 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)

  2. Section 7-1 Review and Preview

  3. 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 z0.025 = 1.96.That is, the critical value z0.025 = 1.96 has an area of 0.025 to its right.

  4. Preview This chapter presents the beginning of inferential statistics. • 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.

  5. Section 7-2 Estimating a Population Proportion

  6. 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. Definition A point estimate is a single value (or point) used to approximate a population parameter.

  8. The sample proportion p is the best point estimate of the population proportion p. Definition ˆ

  9. 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.

  10. 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:

  11. NOTE We will learn how to construct confidence intervals from a sample statistic later using a formula.

  12. 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

  13. 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

  14. 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

  15. 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

  16. page 319, Figure 7-1 Interpreting a Confidence Interval

  17. Know the correct interpretation of a confidence interval. Caution

  18. NEXT We will learn now discuss how to construct confidence intervals from a sample statistic.

  19. 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.

  20. 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

  21. Example Page 328, problem 7  Find z/2for =0.10

  22. Example Page 328, problem 7 ANSWER:

  23. 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 /2in the right tail of the standard normal distribution.

  24. Critical Value Because the standard normal distribution is symmetric about the value of z=0, the value of –z/2is at the vertical boundary for thearea of/2in the left tail

  25. The Critical Value z2 -z/2

  26. 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

  27. Definition Common choices for confidence levels are: 90% confidence level where  = 10%, 95% confidence level where  = 5%, 99% confidence level where  = 1%,

  28. 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  = 5%

  29. z2 for a 95% Confidence Level   z2 = 1.96  = 0.05  

  30. z2 for a 95% Confidence Level z2 -z2 Critical Values  = 5%  2 = 2.5% = .025

  31. 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:

  32. Margin of Error for Proportions NOTE:

  33. Confidence Interval for Estimating a Population Proportion p p = population proportion = 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

  34. 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.

  35. Confidence Interval for Estimating a Population Proportion p ˆ p ˆ p p – E < < + E where

  36. Confidence Interval for Estimating a Population Proportion p ˆ p ˆ p p – E < < + E p + E (p – E, p + E) ˆ ˆ ˆ

  37. Round-Off Rule for Confidence Interval Estimates of p Round the confidence interval limits for p to three significant digits.

  38. Example Page 328 Problem 18:

  39. Example Page 328, problem 18 ANSWER: compute the critical value z/2

  40. Example Page 328, problem 18 ANSWER: compute the sample proportion and

  41. Example Page 328, problem 18 ANSWER: compute the margin of error E

  42. 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 2nd (-) key on bottom row) , then ENTER. This will give z/2

  43. 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/2

  44. 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

  45. 4. Using the value of the calculated margin of error, E and the value of the sample proportion, p, find the values of p – Eand p + E. Substitute those values in the general format for the confidence interval: Procedure for Constructing a Confidence Interval for p - cont ˆ ˆ ˆ ˆ ˆ p – E < p < p + E 5. Round the resulting confidence interval limits to three significant digits.

  46. Example Page 328 Problem 22:

  47. Example Page 328, problem 22 95% confidence interval gives  = 5%, compute the critical value z/2

  48. Example Page 328, problem 22 compute the sample proportion and

  49. Example Page 328, problem 22 compute the margin of error E

  50. Example Page 328, problem 22 compute the upper and lower limits of the confidence interval upper limit lower limit

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