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Chapter 7 Density Curve - A curve that describes the overall pattern of a distribution. - All Density Curves have and area of 1 or 100% -Entire graph mush be above the x-axis -Normal Distribution – Bell Shape N ( , ) -The area under the curve is the proportion of observations that fall into that interval Probability The likelihood or chance that an event will occur ranges from 0 to 1 - The sum of all Outcomes in an experiment is equal to 100% A quantity whose value changes. Today you will learn about a few different types of variables. a variable whose value is obtained by counting Examples: number of students present number of red marbles in a jar number of heads when flipping coins a variable whose value is obtained by measuring Examples: height of students in class weight of text books distance traveled between classes a variable whose value is a numerical outcome of a random phenomenon denoted with a capital letter, X can be discrete or continuous The probability distribution of a random variable X tells what the possible values of X are and how probabilities are assigned to those values A coin is flipped 3 times and the sequence of heads and tails are recorded. The sample space for this experiment is: Let the random variable X be the number of heads in three coin tosses. Thus, X asigns each outcome a number from the set (0, 1, 2, 3). Outcome X HHH HHT HTH THH HTT THT TTH TTT X has a countable number of values. Probability distribution of a discrete random variable X lists the values and their probabilities: Value of X P(X) 0 P 1 The sum of the probabilities is 1 What is the probability distribution of the discrete random variable X that counts the number of heads in four tosses of a coin? The number of heads, X, has possible values 0, 1, 2, 3, 4. These values are not equally likely! P (X = 0) = P(X P( = 2)= X > 1)= P(X = 1) = P(X = 3) = P(X = 4) = P( X 2) 1. P(X < 4) = 2. P( x 2) NC State posts the grade distributions for its courses online. Students in Statistics 101 in fall 2003 semester received 21% A’s, 43% B’s, 30% C’s 5% F’s, and 1% F’s. Choose a Statistics 101 student at random. What is the probability that the student got a B or better? Less than a C? X takes all values in a given interval of numbers The probability distribution of a continuous random variable is shown by a density curve. The probability that X is between an interval of numbers is the area under the density curve between the interval endpoints The probability that a continuous random variable, X is exactly equal to a number is zero REMEMBER: N(μ, σ) is our shorthand for a normal distribution with mean μ and standard deviation σ. So, if X has the N(μ, σ) distribution then the we can standardize using Z X An opinion poll asks an SRS of 1500 American adults what they consider to be the most serious problem facing our schools. Suppose that if we could ask all adults this question 30% say “drugs”. (We will learn about p̂ in the next chapters, so for now go with it…) p̂ 0.3 and N(0.3, 0.0118). What is the probability that the poll results differs from the truth about the population by more than two percentage points? Let “X” represent the sum of two dice A) Define the random variables in this situation. Is it a continuous or discrete random variable? Draw the density curve (ie: probability distribution). X P(X) 2 3 4 5 6 7 8 9 10 11 12 1. P(X > 4) = 2. P( x 2)