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In an election with 4 candidates for one office and 8 candidates for another office, how many different ballots may be printed How many different 4-letter call letters for a student radio station can be made if a. The first three positions must be letters that do not repeat, and the last three positions are digits. A financial advisor gives her client 9 potential investments and asked her to select and rank her top 5. An internet installer is given 12 customers that need installation to be completed by the end of the day. How many samples of 3 parts can be drawn in which 2 are not defective and 1 is defective In a club with 12 varsity and 10 junior varsity members, a 5-member committee will be randomly chosen. Because the student had such a busy schedule, he or she could not study and guesses randomly at each answer. These two examples illustrate two different types of probability problems involving discrete random variables. A random variable is a numerical variable that describes the outcomes of a statistical experiment in words. Random Variable Notation We will use upper case letters such as X or Y to denote a random variable. If X is a random variable, then X is described in words, and the value x is given as a number. Because you can count the possible values that X can take on and the outcomes are random (the x values 0, 1, 2, 3), X is a discrete random variable. If we have a random variable that has only finitely many outcomes, then we can make a table that shows the values x in one column and the corresponding probabilities in another column. In particular, a discrete probability distribution function has two key characteristics: 1. Then the following is the probability distribution for X: x 0 1 2 3 4 5 P(x) 2/50 11/50 23/50 9/50 4/50 1/50 Solution 5. The sum of the probabilities is one, that is, 2 11 23 9 4 1 + + + + + =1 50 50 50 50 50 50 5. For a random sample of 50 patients, the following information was obtained: Why is this a discrete probability distribution Let X = the number of heads in four tosses; so the values of X are x = 0, 1, 2, 3, 4. Then we can represent the sample space using the table: (1, 1) (2, 1) (3, 1) (4, 1) (5, 1) (6, 1) (1, 2) (2, 2) (3, 2) (4, 2) (5, 2) (6, 2) (1, 3) (2, 3) (3, 3) (4, 3) (5, 3) (6, 3) (1, 4) (2, 4) (3, 4) (4, 4) (5, 4) (6, 4) (1, 5) (2, 5) (3, 5) (4, 5) (5, 5) (6, 5) (1,6) (2, 6) (3, 6) (4, 5) (5, 6) (6, 6) Let X = number of dots facing up in a single toss. Using the sample space above, we see that P (x = 2) = table: x P(x) 2 1 36 3 2 36 4 3 36 5 4 36 6 5 36 7 6 36 8 5 36 1 2, P (x = 3) =, etc. That is, if we performed the probability experiment over and over, we would expect the average of the numerical values to be this amount. 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It could be wise to schedule more time since there is an associated 20% chance that the maintenance time will be greater than 1. A confidence interval is another type of estimate but, instead of being just one number, it is an interval of numbers. Construct an approximate 95% confidence interval for the true mean number of meals students eat out each week. To get a 90% confidence interval, we must include the central 90% of the probability of the normal distribution. This can also be found using appropriate commands on other calculators, using a computer, or using a probability table for the standard normal distribution. Use your calculator, a computer, or a probability table for the standard normal distribution to find z0. These intervals are different for several reasons: they were calculated from different samples, the samples were different sizes, and the intervals were calculated for different levels of confidence. Even though the intervals are different, they do not yield conflicting information. Interpretation We estimate with 95% confidence that the true population mean for all statistics exam scores is between 67. Calculating the Sample Size n If researchers desire a specific margin of error, then they can use the error bound formula to calculate the required sample size. Therefore, 217 Foothill College students should be surveyed in order to be 95% confident that we are within two years of the true population mean age of Foothill College students. Just replacing with s did not produce accurate results when he tried to calculate a confidence interval. Random sampling is assumed, but that is a completely separate assumption from normality.  