WARM UP 1. A national investigation reveals that 24% of 16-21 year olds get traffic tickets. You collect a random sample of 200 16-21 year olds. What is the Probability that between 45 and 55 of the 200 have received tickets? Check the Assumptions. Change both 45 and 55 to % and find their two z scores. 2. A national investigation reveals that 16-21 year olds travel an average of 12 miles over the speed limit, σ = 2.8. You collect a random sample of 20 16-21 year olds. What is the Probability they travel between 15 to 20 miles over the speed limit? Check the Assumptions

1. A national investigation reveals that 24% of. 16-21 year olds get 1. A national investigation reveals that 24% of 16-21 year olds get traffic tickets. You collect a random sample of 200 16-21 year olds. What is the Probability that between 22.5% & 27.5% of 16-21 year olds received tickets? Check the Assumptions. 2. A national investigation reveals that 16-21 year olds travel an average of 12 miles over the speed limit, σ = 2.8. You collect a random sample of 20 16-21 year olds. What is the Probability they travel between 15 to 20 miles over the speed limit? Proceed with Caution! PWC

THE CENTRAL LIMIT THEOREM (C.L.T.): Shape The Central Limit Theorem states that the sampling distribution of mean or proportion will be Approximately Normal, regardless of the distribution of the population, as long as the n is Large, & the observations are independent and collected by an SRS. THE LAW OF LARGE NUMBERS: Center As the number of observations, n, increases, the closer and closer the mean gets to the true population mean.

Sampling Distribution of Pennies 2 4 9 14 21 27 66 142 13 1958 1963 1968 1973 1978 1983 1988 1993 1998 2003 2008 150 – 125 – 100 – 75 – 50 – 25 – 0 – Penny’s Date n = 35 n = 25

EXAMPLE: In a large factory batch of M&M’s it is stated that 12% of the candies will be green. You collect a random sample of 86 M&M’s. What is the probability that less than 10% of the candies will be green?

EXAMPLE: The weight of a bag of Potato Chips is supposed to have mean = 20.4 oz with std. dev.= 1.24 oz. The product is declared underweight if it weighs less than 18 oz which is stated on the bag. With a SRS sample of 34 bags find the probability that the Product will be underweight. Collected by an SRS, and Approximately Normal: n>30 -> C.L.T.

HW Page 430: 25(c,d), 36(c,d),37, 38

HW Page 430: 25(c,d), 36(c,d),37, 38

EXAMPLE: The weight of a bag of Potato Chips is supposed to have mean = 20.4 oz with std. dev.= 1.24 oz. The product is declared underweight if it weighs less than 18 oz which is stated on the bag. With a SRS sample of a single bag find the probability that the product is underweight. Assume a Normal Population. With a SRS sample of 34 bags find the probability that the Product will be underweight. Approximately Normal –Stated And Collected by an SRS. Approximately Normal: n>30 -> C.L.T. And Collected by an SRS.

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 = invNorm(.20)= -.8416 20% 22.8 27 31.2 35.4 39.6 43.8 48 x=? x = 31.865”

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? z = invNorm(.75)= .6745 25% 218 234 250 266 282 298 314 x=? x = 276.79

1. The Test over Chapter ninteen has traditionally. had a mean of 86 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 .

With μ = 86.2% and σ = 8.5%. a.) When n = 1 the b.) When n = 16 the c.). 86.2 88.3 92.6 90.5 79.8 82.0 84.1 60.7 69.2 77.7 86.2 94.7 103.2 111.7