Useful Inference Facts Some facts about inference for means and proportions A common format Several types of confidence intervals for population means and population proportions have the following general format. estimate (critical value) (spread) Here, estimate is the value of a statistic used to estimate one of the population parameter(s), spread is the standard deviation or standard error of the statistic, and critical value is a number that depends on the level of confidence and the sampling distribution of the statistic. Several hypothesis tests for population means and population proportions employ a test statistic of the following form. estimate - parameter spread Here, estimate and spread are as above, and parameter is the population parameter of interest. from the population. Notice that the confidence Espaol (b) Press the "Generate Samples" button below to simulate taking 19 more samples of size n = intervals for these samples are drawn automatically. Then complete parts (c) and (d) below the table. 75% 75% 90% 90% x lower upper lower upper limit limit limit limit 75% confidence intervals S1 112.8 ? ? ? ? S2 115.6 112.3 118.9 110.9 120.3 S3 116.5 113.2 119.8 111.8 121.2 $4 113.6 110.3 116.9 108.9 118.3 $5 111.0 107.7 114.3 106.3 115.7 $6 114.6 111.3 117.9 109.9 119.3 57 113.9 110.6 117.2 109.2 118.6 S8 111.0 107.7 114.3 106.3 115.7 S9 113.8 110.5 117.1 109.1 118.5 117.8 111.6 121.0 510 113.1 109.8 1164 | 1084 511 1163 113.0 | 119.6 $12 119.0 115.7 122.3 114.3 123.7 $13 114.0 110.7 117.3 109.3 118.7 S14 119.0 115.7 122.3 114.3 123.7 515 1095 106.2 112.8 1048 114.2 $16 114.5 111.2 117.8 109.8 119.2 S17 113.9 110.6 117.2 109.2 118.6 S18 109.5 106.2 112.8 104.8 114.2 S19 116.6 113.3 119.9 111.9 121.3 $20 116.4 113.1 119.7 111.7 121.1 105.0 127.0 105.0 90% confidence intervals 127.0 ? Ex Specific situations Inference situation Estimating (via a confidence interval) or testing whether equals a certain value Known Normal population or n large Estimating (via a confidence interval) or testing whether equals a certain value Unknown Normal population Estimating p (via a confidence interval) np > 5 and n (1-p) > 5 Estimating statistic Standard deviation or standard error x n S x n p(1-p) P n Testing whether p = Po np >5 and n (1-p) > 5 Estimating (via a confidence interval) or testing whether Known and o Normal populations or n and n large Estimating (via a confidence interval) or testing whether Unknown and o n and n large Estimating (via a confidence interval) or testing whether 41-42 Po(1-Po) P n x1-X2 155 2 02 + n1 2 $1 $2 X1 X2 + n1 - - X1-X2 S 2 'P 1 1 + - n1 n2 Unknown and 2; 0 = 02 01 Normal populations Estimating P1 P2 (via a confidence interval) n P > 5, n(1 P) > 5, P25, and 2(1-P2) >5 Testing whether P = P2 nP > 5, n(1p) > 5, n2P25, and 2(1-P2) >5 (n1)s+ (n-1) s P11-P1 P21-P2) P1 P2 + n1 m P1-12 1 1 p(1 - p) n1 n n1P1+n2P2 Inference situation Some facts about inference for other parameters Estimating (via a confidence interval) or testing whether o equals a certain value Normal populations Estimating statistic Test statistic (m-1) follows a chi-square distribution with n-1 degrees of freedom Estimating 2522 (via a confidence interval) or testing whether o = Normal populations 213 follows an F distribution with n-1 numerator degrees of freedom and n-1 denominator degrees of freedom (b) Press the "Generate Samples" button below to simulate taking 19 more samples of size n = 15 from the population. Notice that the confidence intervals for these samples are drawn automatically. Then complete parts (c) and (d) below the table. x 75% 75% 90% 90% lower upper lower upper S1 112.8 limit limit limit limit ? ? ? 75% confidence intervals ? S2 S3 S4 Generate Samples S5 S6 S7 S8 S9 S10 S11 S12 S13 S14 S15 S16 S17 90% confidence intervals ? B You have taken a random sample of size n = 15 from a normal population that has a population mean of = 115 and a population standard deviation of = 11. Your sample, which is Sample 1 in the table below, has a mean of x = 112.8. (In the table, Sample 1 is written "S1", Sample 2 is written "S2", etc.) (a) Based on Sample 1, graph the 75% and 90% confidence intervals for the population mean. Use 1.150 for the critical value for the 75% confidence interval, and use 1.645 for the critical value for the 90% confidence interval. (If necessary, consult a list of formulas.) Enter the lower and upper limits on the graphs to show each confidence interval. Write your answers with one decimal place. For the points ( and ), enter the population mean, = 115. 105.0 105.0 75% confidence interval 116.0 127.0 105.0 127.0 105.0 90% confidence interval 116.0 127.0 127.0 B S18 S19 S20 18 (c) Notice that for 20 105.0 127.0 105.0 = 90% of the samples, the 90% confidence interval contains the population mean. Choose the 127.0 correct statement. When constructing 90% confidence intervals for 20 samples of the same size from the population, exactly 90% of the samples will contain the population mean. When constructing 90% confidence intervals for 20 samples of the same size from the population, at most 90% of the samples will contain the population mean. When constructing 90% confidence intervals for 20 samples of the same size from the population, it is possible that more or fewer than 90% of the samples will contain the population mean. (d) Choose ALL that are true. The 75% confidence interval for Sample 9 is narrower than the 90% confidence interval for Sample 9. This must be the case, because when a confidence interval is constructed for a sample, the greater the level of confidence, the wider the confidence interval. The 90% confidence interval for Sample 9 indicates that 90% of the Sample 9 data values are between 109.1 and 118.5. From the 75% confidence interval for Sample 9, we know that there is a 75% probability that the population mean is between 110.5 and 117.1. If there were a Sample 21 of size n = 30 taken from the same population as Sample 9, then the 90% confidence interval for Sample 21 would be narrower than the 90% confidence interval for Sample 9. Espaol EX