6-4 Large-Sample Confidence Intervals for the Population Proportion, p

6-4 Large-Sample Confidence Intervals for the Population Proportion, p

Large-Sample Confidence Interval for the Population Proportion, p (Example 6-4) A marketing research firm wants to estimate the share that foreign companies have in the American market for certain products. A random sample of 100 consumers is obtained, and it is found that 34 people in the sample are users of foreign-made products; the rest are users of domestic products. Give a 95% confidence interval for the share of foreign products in this market. Thus, the firm may be 95% confident that foreign manufacturers control anywhere from 24.72% to 43.28% of the market.

Large-Sample Confidence Interval for the Population Proportion, p (Example 6-4) – Using the Template

Reducing the Width of Confidence Intervals - The Value of Information The width of a confidence interval can be reduced only at the price of: a lower level of confidence, or a larger sample. Lower Level of Confidence Larger Sample Size 90% Confidence Interval Sample Size, n = 200

6-5 Confidence Intervals for the Population Variance: The Chi-Square (2) Distribution The sample variance, s2, is an unbiased estimator of the population variance, 2. Confidence intervals for the population variance are based on the chi-square (2) distribution. The chi-square distribution is the probability distribution of the sum of several independent, squared standard normal random variables. The mean of the chi-square distribution is equal to the degrees of freedom parameter, (E[2] = df). The variance of a chi-square is equal to twice the number of degrees of freedom, (V[2] = 2df).

The Chi-Square (2) Distribution The chi-square random variable cannot be negative, so it is bound by zero on the left. The chi-square distribution is skewed to the right. The chi-square distribution approaches a normal as the degrees of freedom increase. C h i - S q u a r e D i s t r i b u t i o n : d f = 1 , d f = 3 , d f = 5 . 1 . 9 df = 10 . 8 . 7 ) . 6 2  df = 30 ( . 5 f . 4 . 3 df = 50 . 2 . 1 . 5 1  2

Values and Probabilities of Chi-Square Distributions Area in Right Tail .995 .990 .975 .950 .900 .100 .050 .025 .010 .005 Area in Left Tail df .005 .010 .025 .050 .100 .900 .950 .975 .990 .995 1 0.0000393 0.000157 0.000982 0.000393 0.0158 2.71 3.84 5.02 6.63 7.88 2 0.0100 0.0201 0.0506 0.103 0.211 4.61 5.99 7.38 9.21 10.60 3 0.0717 0.115 0.216 0.352 0.584 6.25 7.81 9.35 11.34 12.84 4 0.207 0.297 0.484 0.711 1.06 7.78 9.49 11.14 13.28 14.86 5 0.412 0.554 0.831 1.15 1.61 9.24 11.07 12.83 15.09 16.75 6 0.676 0.872 1.24 1.64 2.20 10.64 12.59 14.45 16.81 18.55 7 0.989 1.24 1.69 2.17 2.83 12.02 14.07 16.01 18.48 20.28 8 1.34 1.65 2.18 2.73 3.49 13.36 15.51 17.53 20.09 21.95 9 1.73 2.09 2.70 3.33 4.17 14.68 16.92 19.02 21.67 23.59 10 2.16 2.56 3.25 3.94 4.87 15.99 18.31 20.48 23.21 25.19 11 2.60 3.05 3.82 4.57 5.58 17.28 19.68 21.92 24.72 26.76 12 3.07 3.57 4.40 5.23 6.30 18.55 21.03 23.34 26.22 28.30 13 3.57 4.11 5.01 5.89 7.04 19.81 22.36 24.74 27.69 29.82 14 4.07 4.66 5.63 6.57 7.79 21.06 23.68 26.12 29.14 31.32 15 4.60 5.23 6.26 7.26 8.55 22.31 25.00 27.49 30.58 32.80 16 5.14 5.81 6.91 7.96 9.31 23.54 26.30 28.85 32.00 34.27 17 5.70 6.41 7.56 8.67 10.09 24.77 27.59 30.19 33.41 35.72 18 6.26 7.01 8.23 9.39 10.86 25.99 28.87 31.53 34.81 37.16 19 6.84 7.63 8.91 10.12 11.65 27.20 30.14 32.85 36.19 38.58 20 7.43 8.26 9.59 10.85 12.44 28.41 31.41 34.17 37.57 40.00 21 8.03 8.90 10.28 11.59 13.24 29.62 32.67 35.48 38.93 41.40 22 8.64 9.54 10.98 12.34 14.04 30.81 33.92 36.78 40.29 42.80 23 9.26 10.20 11.69 13.09 14.85 32.01 35.17 38.08 41.64 44.18 24 9.89 10.86 12.40 13.85 15.66 33.20 36.42 39.36 42.98 45.56 25 10.52 11.52 13.12 14.61 16.47 34.38 37.65 40.65 44.31 46.93 26 11.16 12.20 13.84 15.38 17.29 35.56 38.89 41.92 45.64 48.29 27 11.81 12.88 14.57 16.15 18.11 36.74 40.11 43.19 46.96 49.65 28 12.46 13.56 15.31 16.93 18.94 37.92 41.34 44.46 48.28 50.99 29 13.12 14.26 16.05 17.71 19.77 39.09 42.56 45.72 49.59 52.34 30 13.79 14.95 16.79 18.49 20.60 40.26 43.77 46.98 50.89 53.67

Template with Values and Probabilities of Chi-Square Distributions

Confidence Interval for the Population Variance A (1-)100% confidence interval for the population variance * (where the population is assumed normal) is: where is the value of the chi-square distribution with n - 1 degrees of freedom that cuts off an area to its right and is the value of the distribution that cuts off an area of to its left (equivalently, an area of to its right). * Note: Because the chi-square distribution is skewed, the confidence interval for the population variance is not symmetric

Confidence Interval for the Population Variance - Example 6-5 In an automated process, a machine fills cans of coffee. If the average amount filled is different from what it should be, the machine may be adjusted to correct the mean. If the variance of the filling process is too high, however, the machine is out of control and needs to be repaired. Therefore, from time to time regular checks of the variance of the filling process are made. This is done by randomly sampling filled cans, measuring their amounts, and computing the sample variance. A random sample of 30 cans gives an estimate s2 = 18,540. Give a 95% confidence interval for the population variance, 2.

Example 6-5 (continued) C h i - S q u a r e D s t b o n : d = 9 Area in Right Tail df .995 .990 .975 .950 .900 .100 .050 .025 .010 .005 . . . . . . . . . . . 28 12.46 13.56 15.31 16.93 18.94 37.92 41.34 44.46 48.28 50.99 29 13.12 14.26 16.05 17.71 19.77 39.09 42.56 45.72 49.59 52.34 30 13.79 14.95 16.79 18.49 20.60 40.26 43.77 46.98 50.89 53.67 7 6 5 4 3 2 1 .  f ( ) C h i - S q u a r e D s t b o n : d = 9 0.025 0.95

Example 6-5 Using the Template Using Data

6-6 Sample-Size Determination Before determining the necessary sample size, three questions must be answered: How close do you want your sample estimate to be to the unknown parameter? (What is the desired bound, B?) What do you want the desired confidence level (1-) to be so that the distance between your estimate and the parameter is less than or equal to B? What is your estimate of the variance (or standard deviation) of the population in question? } Bound, B

Sample Size and Standard Error The sample size determines the bound of a statistic, since the standard error of a statistic shrinks as the sample size increases:  Standard error of statistic Sample size = n Sample size = 2n

Minimum Sample Size: Mean and Proportion Minimum re quired sam ple size i n estimati ng the pop ulation mean, : Bound of e stimate: B = z 2 m s a n proportion , p $ pq

Sample-Size Determination: Example 6-6 A marketing research firm wants to conduct a survey to estimate the average amount spent on entertainment by each person visiting a popular resort. The people who plan the survey would like to determine the average amount spent by all people visiting the resort to within $120, with 95% confidence. From past operation of the resort, an estimate of the population standard deviation is s = $400. What is the minimum required sample size?

Sample-Size for Proportion: Example 6-7 The manufacturers of a sports car want to estimate the proportion of people in a given income bracket who are interested in the model. The company wants to know the population proportion, p, to within 0.01 with 99% confidence. Current company records indicate that the proportion p may be around 0.25. What is the minimum required sample size for this survey?

6-7 The Templates – Optimizing Population Mean Estimates

6-7 The Templates – Optimizing Population Proportion Estimates