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1 Lecture 6 Outline 1. Random Variables a. Discrete Random Variables b. Continuous Random Variables 2. Symmetric Distributions 3. Normal Distributions 4. The Standard Normal Distribution
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2 Lecture 6 1. Random Variables Two kinds of random variables: a. Discrete (DRV) Outcomes have countable values Possible values can be listed E.g., # of people in this room Possible values can be listed: might be …28 or 29 or 30…
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3 Lecture 6 1. Random Variables Two kinds of random variables: b. Continuous (CRV) Not countable Consists of points in an interval E.g., time till coffee break
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4 Lecture 6 1. Random Variables The form of the probability distribution for a CRV is a smooth curve. Such a distribution may also be called a Frequency Distribution Probability Density Function
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5 Lecture 6 1. Random Variables In the graph of a CRV, the X axis is whatever you are measuring (e.g., exam scores, depression scores, # of widgets produced per hour). The Y axis measures the frequency of scores.
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6 Lecture 6 X The Y-axis measures frequency. It is usually not shown.
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7 Lecture 6 2. Symmetric Distributions In a symmetric CRV, 50% of the area under the curve is in each half of the distribution. P(x ≤ ) = P(x ≥ ) =.5 Note: Because points are infinitely thin, we can only measure the probability of intervals of X values – not of individual X values.
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8 Lecture 6 µ 50% of area
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9 Lecture 6 3. Normal Distributions A particularly important set of CRVs have probability distributions of a particular shape: mound-shaped and symmetric. These are “normal distributions” Many naturally-occurring variables are normally distributed.
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10 Lecture 6 Normal Distributions are perfectly symmetrical around their mean, . have the standard deviation, , which measures the “spread” of a distribution – an index of variability around the mean.
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11 Lecture 6 µ
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12 Lecture 6 Standard Normal Distribution The area under the curve between and some value X ≥ has been calculated for the “standard normal distribution” and is given in the Z table (Table IV). E.g., for Z = 1.62, area =.4474 (Note that for the mean, Z = 0.)
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13 Lecture 6 XZ = 1.62 Z = 0 Area gives the probability of finding a score between the mean and X when you make an observation.4474
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14 Lecture 6 Using the Standard Normal Distribution Suppose average height for Canadian women is 160 cm, with = 15 cm. What is the probability that the next Canadian woman we meet is more than 175 cm tall? Note that this is a question about a single case and that it specifies an interval.
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15 Lecture 6 Using the Standard Normal Distribution 160175 We need this areaTable gives this area
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16 Lecture 6 Remember that area above the mean, , is half (.5) of the distribution. µ
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17 Lecture 6 Using the Standard Normal Distribution 160175 Call this shaded area P. We can get P from Table IV
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18 Lecture 6 Using the Standard Normal Distribution Z = X - = 175-160 15 = 1.00 Now, look up Z = 1.00 in the table. Corresponding area (= probability) is P =.3413.
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19 Lecture 6 Using the Standard Normal Distribution 160175 This area is.3413 So this area must be.5 –.3413 =.1587
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20 Lecture 6 Using the Standard Normal Distribution Z = 0Z = 1.0 This area is.3413 So this area must be.5 –.3413 =.1587
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21 Lecture 6 Using the Standard Normal Distribution What is the probability that the next Canadian woman we meet is more than 175 cm tall? Answer:.1587
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22 Lecture 6 Review Area under curve gives probability of finding X in a given interval. Area under the curve for Standard Normal Distribution is given in Table IV. For area under the curve for other normally- distributed variables first compute: Z = X - Then look up Z in Table IV.
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