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WARM UP A statewide poll reveals the only 18% of Texas household participate in giving out Halloween Treats. To examine this you collect a Random Sample of 120 Texas households. 1. What is the Probability that less than 12% of Texas households give out Treats? P pˆ .12 ? P z # ? P z 1.7108 ? z pˆ p p 1 p n .12 .18 P z .18 1 .18 120 normalcdf ( E 99, 1.7108) 0.0436 2. Comment on the validity of your results by checking the Rule of Thumb assumptions. RULE OF THUMB #1: Pop. of Texas Households ≥ 10 • 120 RULE OF THUMB #2: 120 • .18 ≥ 10 AND 120 • (1 – .18) ≥ 10 3. Draw the Sampling Distribution curve. ? N p, p (1 p ) n N(0.18, 0.035) .075 .11 .12 .145 .18 .215 .25 .285 Chapter 18 (continued) Sample Proportion, p̂ use Categorical Data Data collected by COUNTING. Sample Means, x use Quantitative Data Data collected by AVERAGING SAMPLE MEANS (Chapter 18 continued) I. THE BASICS The unknown Population Mean is a parameter with the symbol of: . The Population Standard Deviation has the symbol: The Sample Mean is a statistic with the symbol of: x and the Sample Standard Deviation has the formula s . The Test Statistic is: z x n n ASSUMPTIONS for Sample Means: x N , n #1: The sample was collected using an SRS. #2: Approximate Normal: 1. Large n = (n ≥ 30), OR 2. Stated that its Normal, OR 3. You construct an approximately Normal Histogram/Boxplot from the given data. It is known that each individual Halloween Treat given out has about 120 calories with standard deviation of 50. To examine this you collect a Random Sample of 40 Halloween treats. 1. What is the Probability that the sample average is less than 100 calories? P x 100 ? z x P z 2.5298 ? n 100 120 P z ? 50 40 normalcdf ( E 99, z) 0.0057 2. Comment on the validity of your results by checking the conditons/assumptions. Stated to be a Random sample Approximately Normal due to the Large n. 3. Draw the Sampling Distribution curve. II. THE CALCULATIONS EXAMPLE: A national political poll asked a SRS of 50 adults to give an approval rating of President Obama’s Job in office using a scale from 1(Low) to 10(High). Suppose in fact, nationally Obama’s rating has a Mean of 3.5 and Standard Deviation of 1.2 1. Find the Mean and Standard Deviation of the Sampling Distribution of x . Mean = 3.5 Std. Dev. = .1697 1.2 Std . Dev. 50 2. Explain why you can assume normality in the sample. #1: The sample was collected by an SRS #2: Approximately Normal b/c Large n II. THE CALCULATIONS of PROBABILITY SAME EXAMPLE: A national political poll asked a SRS of 50 adults to give an approval rating of President Obama’s Job in office using a scale from 1(Low) to 10(High). Suppose in fact, nationally Obama’s rating has a Mean of 3.5 and Standard Deviation of 1.2 3. Find the probability Obama has a rating over 4.0. x P x 4.0 ? z Pz # ? n 4.0 3.5 P z P z 2.9463 ? 1.2 50 Probability normalcdf (2.9463, E 99) 0.0016 SAME EXAMPLE: A national political poll asked a SRS of 50 adults to give an approval rating of President Obama’s Job in office using a scale from 1(Low) to 10(High). Suppose in fact, nationally Obama’s rating has a Mean of 3.5 and Standard Deviation of 1.2 4. Find the probability Obama’s approval rating is between 3.2 and 4.0. P 3.2 x 4.0 ? 3.2 3.5 4.0 3.5 P ? 1.2 1.2 50 50 z z x n P 1.7678 z 2.9463 ? Prob.=normalcdf ( 1.7678, 2.9463) 0.9598 Finding Sample Mean Observations, (x) from Probabilities EXAMPLE: Assume that the duration of human pregnancies is Appr. Normal with mean: 266 days and Std Dev.: 16. At least how many days should the longest 25% of all pregnancies last? x z 1 z = invNorm(.75)= .6745 25% 218 234 250 266 282 298 314 x=? x 266 0.6745 16 x = 276.79 Finding Sample Mean Observations, (x) from Probabilities EXAMPLE: Assume that the rainfall in Ithaca, NY is Appr. Normal with mean: 35.4 inches and Std Dev.: 4.2”. Less than how much rain falls in the driest 20% of all years z x z = invNorm(.20)= -.8416 20% 22.8 27 31.2 35.4 39.6 43.8 48 x=? x 35.4 .8416 4.2 x = 31.865” 1. The Test over Chapter ninteen has traditionally had a mean of 86.2% with a std. deviation of 8.5%. Assume the data follows a Normal distribution. a.) If a student is selected at random, what is the probability that he or she will score above 90%? b.) If 16 students are selected at random, what is the probability that their average will be above 90%? c.) Draw the Sampling Distribution Normal Curves for both a and b. On both graphs label all six standard deviations (± 3s) for the N , . n With μ = 86.2% and σ = 8.5%. P a.) When n = 1 the P x 90 ? 1.Appr. Normal 69.2 77.7 P z 90 86.2 ? 8.5 16 P z 1.7882 ? normalcdf (1.7882, E 99) 0.0369 Stated . and 2.Data Collected Randomly SRS 8.5 N 86.2, 16 N 86.2, 2.125 c.).N 86.2, 8.5 1 N 86.2,8.5 60.7 P z 0.4471 ? normalcdf (0.4471, E 99) 0.3274 Stated . and 2.Data Collected Randomly SRS b.) When n = 16 the P x 90 ? 1.Appr. Normal z 90 86.2 ? 8.5 1 86.2 94.7 103.2 111.7