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Chapter 8

Chapter 8. Estimation. Table of Contents. 8.1 Estimating µ When σ Is Known. 8.2 Estimating µ When σ Is Unknown. 8.3 Estimating p in the Binomial Distribution. 8.4 Estimating µ 1 − µ 2 and p 1 − p 2. 8.1 Estimating µ When σ Is Known.

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Chapter 8

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  1. Chapter 8 Estimation

  2. Table of Contents 8.1 Estimating µ When σ Is Known 8.2 Estimating µ When σ Is Unknown 8.3 Estimating p in the Binomial Distribution 8.4 Estimating µ1−µ2 and p1−p2

  3. 8.1 Estimating µ When σ Is Known Assumptions about the random variable x 1. We have a simple random sample of size n drawn from a population of x values. 2. The value of σ, the population standard deviation of x, is known. 3. If the x distribution is normal, then our methods work for any sample size n. 4. If x has an unknown distribution, then we require a sample size n≥ 30. However, if the x distribution is distinctly skewed and definitely not mound-shaped, a sample of size 50 or even 100 or higher may be necessary.

  4. A point estimate of a population parameter is an estimate of the parameter using a single number. x is the point estimate for µ. ¯ When using x as a point estimate for µ, the margin of error is the magnitude of x – µ or |x – µ|. ¯ ¯ ¯ Definitions 8.1 Estimating µ when σ Is Known

  5. Definition For a confidence level c, the critical value zc is the number such that the area under the standard normal curve between –zc and zc equals c. In mathematical notation, zc is the value that satisfies P(–zc < z < zc) = c 8.1 Estimating µ when σ Is Known

  6. Note We have gone over this concept in section 6.3 when we discussed the invNorm() function with the center case. As a refresher, refer to example 9 and figure 6-28 on page 263. With the center case, we used invNorm((1–A)/2). 8.1 Estimating µ when σ Is Known

  7. 8.1 Estimating µ when σ Is Known

  8. The maximal margin of error of |x – µ| at a confidence level of c is defined as ¯ σ E = zc ─ √n Definition 8.1 Estimating µ when σ Is Known

  9. µ σ σ E = zc E = zc ─ ─ √n √n Knowing σ, but not knowing µ, what we’re saying is that there is a c chance that |x – µ| < E. ¯ Area = c 8.1 Estimating µ when σ Is Known

  10. Since there is a c chance that |x – µ| < E, we can rewrite this as –E < x – µ < E, which is what we find in the diagram above. ¯ ¯ Area = c µ σ σ E = zc E = zc ─ ─ √n √n 8.1 Estimating µ when σ Is Known

  11. Since there is a c chance that |x – µ| < E, we can rewrite this as –E < x – µ < E, which is what we find in the diagram above. ¯ ¯ Using an algebra trick, we can also rewrite the above inequality to state that, since there is a c chance that |x – µ| < E, there is also a c chance that x – E < µ < x + E, which is the real goal of all this. ¯ ¯ ¯ 8.1 Estimating µ when σ Is Known

  12. Definition A c confidence interval for µ is an interval computed from sample data in such a way that c is the probability of generating an interval containing the actual value of µ. 8.1 Estimating µ when σ Is Known

  13. Let x be a random variable appropriate to your application. Obtain a simple random sample (of size n) of x values from which you compute the sample mean x. The value of σ is already known (perhaps from a previous study). ¯ How to find a confidence interval for µ when σ is known If you can assume that x has a normal distribution, then any sample size n will work. If you cannot assume this, then use a sample size of n≥ 30. 8.1 Estimating µ when σ Is Known

  14. x – E < µ < x + E ¯ ¯ σ E = zc ─ where x = sample mean of a simple random sample ¯ √n How to find a confidence interval for µ when σ is known Confidence interval for µ when σ is known c = confidence level (0 < c < 1) zc= critical value for confidence level c 8.1 Estimating µ when σ Is Known

  15. Note Two consequences of these formulas are: 1. As c increases, so does E, and the confidence interval widens. 2. As n increases, E decreases, so the confidence interval narrows. 8.1 Estimating µ when σ Is Known

  16. This returns a confidence interval (low x, high x). E, if needed, can be back-calculated by finding x – low x. ¯ Calculator function The TI-83/84 calculator has a function to compute the confidence interval when σ is known. STAT|TESTS|ZInterval… 8.1 Estimating µ when σ Is Known

  17. x = 15.60 min ¯ ZInterval Inpt:Data σ:1.80 x:15.60 n:90 C-Level:.95 Calculate ZInterval (15.228,15.972) x=15.6 n=90 Stats ¯ ¯ If we were asked to find E, it would be E = x – low x = 15.60 – 15.23 = 0.37. ¯ Example 2 σ = 1.80 min n = 90 c = 95% We are 95% confident that 15.23 < µ < 15.97. 8.1 Estimating µ when σ Is Known

  18. x – E to x + E ¯ ¯ Figure 8-4 0.90 Confidence Intervals for Samples of the Same Size At 90%, we have a 1/10 chance that our interval does not contain µ. 8.1 Estimating µ when σ Is Known

  19. Sample size for estimating the mean μ Sometimes we will encounter situations where we don’t want the margin of error E to exceed a certain value, given some confidence level. Remember from earlier that the confidence interval narrows as n increases. The goal is to find the sample size n so that the margin of error E doesn’t exceed a certain value given some confidence level c. 8.1 Estimating µ when σ Is Known

  20. Assuming the distribution of means x is approximately normal, then ¯ ( ) zcσ ² n = E How to find the sample size n for estimating μ when σ is known If n is not a whole number, increase n to the next higher whole number. Note that n is the minimal sample size for a specified confidence level and maximal error of estimate E. 8.1 Estimating µ when σ Is Known

  21. Assuming the distribution of means x is approximately normal, then ¯ ( ) zcσ ² n = E How to find the sample size n for estimating μ when σ is known For this equation, either σ is known, or n ≥ 30 and we can use s as an estimate for σ. 8.1 Estimating µ when σ Is Known

  22. … As a preliminary study, a random sample of 50 freshly caught salmon is weighed. The sample standard deviation of weights of these 50 fish is s ≈ 2.15 lb. How large a sample should be taken to be 99% confident that the sample mean x is within 0.20 lb of the true mean weight µ? ¯ Example 3 We need to find n big enough to reduce E down to 0.20 lb at the 99% confidence level. We also need a value for σ. Since n = 50 ≥ 30 in the preliminary study, we can use σ ≈ s ≈ 2.15 lb. 8.1 Estimating µ when σ Is Known

  23. ( ) ( ) zcσ ² (2.58)(2.15) ² n = = E 0.20 Example 3 = 769.2; n = 770 We need to find n big enough to reduce E down to 0.20 lb at the 99% confidence level. We also need a value for σ. Since n = 50 ≥ 30 in the preliminary study, we can use σ ≈ s = 2.15 lb. 8.1 Estimating µ when σ Is Known

  24. 8.2 Estimating µ When σ Is Unknown In 8.1, we knew σ before trying to determine a confidence interval. Then, if x were normal, any sample size n would work, and if x were not normal, then we would need a sample size of at least 30. In 8.2, we won’t know σ before we try to determine a confidence interval. In this case, we requirex to be normal, or at least approximately normal, or that we have n ≥ 30.

  25. Assume that x has a normal distribution with mean µ. For samples of size n with sample mean x and sample standard deviation s, the t variable ¯ x – µ ¯ t = ─ s/√n Definition has a Student’s t distribution with degrees of freedom d.f. = n – 1. 8.2 Estimating µ when σ Is Unknown

  26. Normal d.f. = 20 d.f. = 8 d.f. = 4 d.f. = 1 Thicker tails than normal –4 –3 –2 –1 0 1 2 3 4 8.2 Estimating µ when σ Is Unknown

  27. Figure 8-6 Area = c –tc 0 tc The critical value tc for a c confidence level is found in a fashion similar to finding zc in that the area above the t-interval [–tc,tc] is c. 8.2 Estimating µ when σ Is Unknown

  28. Figure 8-6 Area = c –tc 0 tc In mathematical notation, tc is the value that satisfies P(–tc < t < tc) = c 8.2 Estimating µ when σ Is Unknown

  29. Degrees of freedom for the t distribution For confidence intervals involving μ, the degrees of freedom that determine the t distribution is one less than the sample size. In other words, d.f. = n − 1 8.2 Estimating µ when σ Is Unknown

  30. Example 4 d.f. = n – 1 = 5 – 1 = 4 c = 0.99 tc = 4.604 8.2 Estimating µ when σ Is Unknown

  31. Let x be a random variable appropriate to your application. Obtain a simple random sample (of size n) of x values from which you compute the sample mean x and the sample standard deviation s. ¯ How to find a confidence interval for µ when σ is unknown If you can assume that x has a normal distribution or simply a mound-shaped symmetric distribution, then any sample size n will work. If you cannot assume this, then use a sample size of n≥ 30. 8.2 Estimating µ when σ Is Unknown

  32. x – E < µ < x + E ¯ ¯ where x = sample mean of a simple random sample ¯ s E = tc ─ √n How to find a confidence interval for µ when σ is unknown Confidence interval for µ when σ is unknown c = confidence level (0 < c < 1) tc= critical value for confidence level c and degrees of freedom d.f. = n – 1 8.2 Estimating µ when σ Is Unknown

  33. This returns a confidence interval (low x, high x). E, if needed, can be back-calculated by finding x – low x. ¯ Calculator function The TI-83Plus calculator has a function to compute the confidence interval when σ is unknown. STAT|TESTS|TInterval… 8.2 Estimating µ when σ Is Unknown

  34. TInterval (44.475,47.81) x=46.14285714 Sx=1.190038015 n=7 TInterval Inpt: Stats List:L1 Freq:1 C-Level:.99 Calculate Data ¯ If we were asked to find E, it would be E = x – low x = 46.143 – 44.475 = 1.668. ¯ Example 4 If we have raw data to start with, we can enter the data into a list, and then choose ‘Data’ for the Input type. Let’s enter data into L1. Freq will always be 1 We are 99% confident that 44.5 < µ < 47.8. 8.2 Estimating µ when σ Is Unknown

  35. ¯ Which distribution should you use for x? Examine problem statement σ is known σ is unknown Use ZInterval Use TInterval 8.2 Estimating µ when σ Is Unknown

  36. r p̂ = n 8.3 Estimating p in the Binomial Distribution In order to estimate p from a sample, we want to have np > 5 and nq > 5, just as with section 6.4. Definition The point estimates for p and q are q̂ = 1 – p̂ where n = number of trials and r = number of successes.

  37. Calculator function The TI-83/84 calculator has a function to compute the confidence interval for a proportion p. STAT|TESTS|1-PropZInt… This returns a confidence interval (low p, high p). E, if needed, can be back-calculated by finding p̂ – low p. 8.3 Estimating p in the Binomial Distribution

  38. 1-PropZInt (.71057,.78943) p=.75 n=800 ˆ Example 6 r = 600 n = 800 (c) Successes = 600 > 5, failures = 200 > 5. Yes, a normal approximation is justified. (d) 1-PropZInt x:600 n:800 C-Level:.99 Calculate r becomes x in the TI-83/84 We are 99% confident that 0.71 < p < 0.79. If we were asked to find E, it would be E = p̂ – low p = 0.75 – 0.7106 = 0.0394. 8.3 Estimating p in the Binomial Distribution

  39. General interpretation of poll results 1. When a poll states the results of a survey, the proportion reported to respond in the designated manner is p̂, the sample estimate of the population proportion. 2. The margin of error is the maximal error E of a 95% confidence interval for p. 3. A 95% confidence interval for the population proportion p is poll report p̂ – margin of error E < p < poll report p̂ + margin of error E 8.3 Estimating p in the Binomial Distribution

  40. ( ) zc ² n = p(1 – p) E ( ) 1 zc ² n = 4 E Definition For a binomial distribution If we have a preliminary estimate for p, then If we have no preliminary estimate for p, then In both cases, if n is not a whole number, then always round up. 8.3 Estimating p in the Binomial Distribution

  41. ( ) ( ) 1 1 zc ² 1.96 ² n = = 4 4 E 0.01 ( ) ( ) zc ² 1.96 ² n = p(1 – p) = (0.86)(1 – 0.86) E 0.01 Example 7 E = 0.01 = 9604 (a) (b) p ≈ 0.86 = 4625.2864; n = 4626 8.3 Estimating p in the Binomial Distribution

  42. 8.4 Estimating µ1 − µ2 and p1 − p2 Definitions Two samples are independent if sample data drawn from one population are completely unrelated to the selection of sample data from the other population. Two samples are dependent if each data value in one sample can be paired with a corresponding data value in the other sample.

  43. Note 8.4 deals exclusively with data from independent samples. 8.5 Estimating µ1−µ2 and p1−p2

  44. Meaning of figure 8-7 Back in section 7.1, we studied an example of trout lengths from a pond. 8.5 Estimating µ1−µ2 and p1−p2

  45. We took n values per sample from the x distribution, and a new distribution, the x distribution of size n, was created. ¯ Meaning of figure 8-7 8.5 Estimating µ1−µ2 and p1−p2

  46. Meaning of figure 8-7 Now suppose there is another set of data to examine, let’s say from a nearby pond. To distinguish these data, let’s call one of them x1 and the other x2. 8.5 Estimating µ1−µ2 and p1−p2

  47. If we randomly select one data value from each of the x1 and x2 populations, we can take their difference and get a new data value x1 – x2. ¯ ¯ ¯ ¯ Meaning of figure 8-7 = –1.7 10.6 – 12.3 8.5 Estimating µ1−µ2 and p1−p2

  48. Meaning of figure 8-7 We can now make an x1 – x2 distribution. ¯ ¯ 10.8 – 13.2 = –2.4 10.4 – 11.9 = –1.5 –1.7 8.5 Estimating µ1−µ2 and p1−p2

  49. Let x1 and x2 have normal distributions with means µ1 and µ2 and standard deviations σ1 and σ2, respectively. If we take independent random samples of size n1 from the x1 distribution and of size n2 from the x2 distribution, then the variable x1 – x2 has ¯ ¯ ─────── √ ² ² σ1 σ2 + n1 n2 Theorem 8.1 1. a normal distribution 2. mean µ1 – µ2 3. standard deviation 8.5 Estimating µ1−µ2 and p1−p2

  50. Calculator function The TI-83/84 calculator has a function to compute the confidence interval for µ1 – µ2 when both σ1 and σ2 are known. STAT|TESTS|2-SampZInt… 8.5 Estimating µ1−µ2 and p1−p2

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