Favor Oppose Total Republican 299 98 397 Democrat 77 171 248 Other 118 87 205 494 356 850 Which of the following does not support the conclusion that being a Republican and favoring the death penalty are not independent? (a) 299/249 not equal to 98/356 (b) 299/494 not equal to 397/850 (c) 494/850 not equal to 299/397 (d) 299/397 not equal to 77/248 (e) (397)(494)/850 not equal to 299

Random Variables Review

M&Ms According to Mars candy co, 20% of its plain M&M candies are orange. Assume that this claim is true. Suppose you reach into a large bag of M&Ms without looking and pull out 8 candies. Let X= # of orange Find and interpret the mean and stdev Would you be surprised if none of the candies were orange? Compute an appropriate prob to support your answer

Random Variables take on numeric values that describe outcomes of a chance process. Probability distributions are all possible values of a random variable and their probabilities.

All Probability Distributions Discrete Continuous Normal Binomial Geometric

(recorded for 2 million babies born in a single year) A “John Doe” Example Baby Apgar Scores (recorded for 2 million babies born in a single year) Score 1 2 3 4 5 6 7 8 9 10 Prob .001 .006 .007 .008 .012 .020 .038 .099 .319 .437 .053 Legit? Prob that a baby is born healthy? (7 or higher) Describe the distribution―SOCS

For Discrete Prob Distributions: Baby Apgar Scores Score 1 2 3 4 5 6 7 8 9 10 Prob .001 .006 .007 .008 .012 .020 .038 .099 .319 .437 .053 For Discrete Prob Distributions:

(8.128) (2.066) Baby Apgar Scores 1 2 3 4 5 6 7 8 9 10 Score Prob .001 1 2 3 4 5 6 7 8 9 10 Prob .001 .006 .007 .008 .012 .020 .038 .099 .319 .437 .053 (8.128) (2.066)

Baby Apgar Scores Score 1 2 3 4 5 6 7 8 9 10 Prob .001 .006 .007 .008 .012 .020 .038 .099 .319 .437 .053 What it the probability that a randomly selected infant is within one standard deviation of the mean?

Binomial Probability Distributions Binary—only options, Success & Failure Trials independent of each other Fixed number of trials (n) P(Success) = p If all these conditions are met, then we can abbreviate: B(n, p)

Binomial Probability Distributions

Binomial Prob Distributions Example In a family, each of the children has a 0.25 chance of having type O blood. The parents in one such family had 5 children. Probability that exactly 3 are type O? Should the parents be surprised if more than 3 are type O?

Geom Prob Distributions Example In a family, each of the children has a 0.25 chance of having type O blood. Imagine that the parents decide to have children UNTIL a child with type O is born. Probability that the parents will have 2 children? Probability that the parents will have more than 3 children?

Random Variables: Transformations Let X be a random variable with: Mean = 𝜇 𝑋 and Stdev = 𝜎 𝑋 Y is found by taking every value of X, multiplying it by some value and adding some value: Y = a + bX Y is a random variable with: Mean = a +b 𝜇 𝑋 StDev =𝑏 𝜎 𝑋

Random Variables: Transformations If X & Y are random variables: 𝜇 𝑋+𝑌 = 𝜇 𝑋 + 𝜇 𝑌 𝜇 𝑋−𝑌 = 𝜇 𝑋 − 𝜇 𝑌 If X & Y are independent random vars: 𝜎 2 𝑋+𝑌 = 𝜎 2 𝑋 + 𝜎 2 𝑌 𝜎 2 𝑋−𝑌 = 𝜎 2 𝑋 + 𝜎 2 𝑌 If X and Y are indep. & each approx. normally distributed then the random variable X±Y will also be approx. normally distributed.

Transformations Example 1: A statistics book has an approximately norm distribution of weight with mean of 4.150 lbs. and stdev of 0.032 lbs. It will come packaged in a box that has a weight also approximately norm distributed with mean of 0.833 lbs. and stdev of 0.021 lbs. If books are shipped 8 per box, what is the mean and stdev of a box of these books?

Transformations Example 2 The weight of US men aged 19-39 is approximately normally distributed with mean of 194.7 lbs, stdev of 8.3 lbs. Likewise for women: mean of 131.4 lbs with mean of 7.5 lbs. 6 men and 6 women get on an elevator with a max weight stated to be 2000 lbs. What is the probability that the elevator will be over its maximum carrying weight?

Example 2 How would your answer change if you found out that this was an express elevator exclusively serving the marriage license department in a local government facility?

The human body temperature is approximately normally distributed with a mean of 98.2°F and standard deviation of 0.62°F. Use the conversion of C = 0.556(F) – 17.778 and find the mean and standard deviation of this distribution in Celsius.

A high school exit test based on the national Common Core Standards is being examined to see if scores can predict college success. Students take tests in four areas: Math, English, Science and Social Studies. The composite score is the sum of the area scores. The scores in each area are approximately normally distributed.

a) What is the probability that a randomly selected student will score above 600 on the Science portion of this high school exit test? b) The composite score is the sum of the area scores. What is the mean and standard deviation of the composite score? c) One teacher wants to help tutor students that have a score of less than 2000. Do you think it likely that she will have students to tutor? Justify your answer.

What four conditions define a binomial random variable? What four conditions define a geometric random variable?

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