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P x 06826 so 6826 of all possible observed

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Unformatted text preview: σ The probability that x could take any value in the range between two given values a and b (a < b) is P(a ≤ x ≤ b) P(a ≤ x ≤ b) is the is area colored in blue area under the normal curve and between the values the x = a and x = b 19 Three Important Areas under the Normal Curve 1. P (µ – σ ≤ x ≤ µ + σ) = 0.6826 So 68.26% of all possible observed values of x are within (plus or minus) one standard deviation of µ 2. P (µ – 2σ ≤ x ≤ µ + 2σ) = 0.9544 So 95.44% of all possible observed values of x are within (plus or minus) two standard deviations of µ 3. P (µ – 3σ ≤ x ≤ µ + 3σ) = 0.9973 So 99.73% of all possible observed values of x are within (plus or minus) three standard deviations of µ 20 Three Important Areas under the Normal Curve (Visually) The Empirical Rule for Normal Populations 21 The Standard Normal Distribution If x is normally distributed with mean µ and standard deviation σ, then the random variable z x −µ z= σ is normally distributed with mean 0 and standard deviation 1; this normal is called the standard normal distribution. normal 22 The Standard Normal Distribution z measures the number of standard deviations that x is from the mean µ • The algebraic sign on z indicates on which side of µ is x • z is positive if x > µ (x is to the right of µ on the number line) • z is negative if x < µ (x is to the left of µ on the number line) 23 The Standard Normal Table Page 860-861 The standard normal table is a table that lists the The cumulative areas under the standard normal curve. This table is very important. Always look at the accompanying figure for Always accompanying guidance on how to use the table guidance 24 The Standard Normal Table The values of z (accurate to the nearest The tenth) in the table range from -3.99 to 3.99 in increments of 0.01 in The areas under the normal curve to the left The of any value of z are given in the body of the table the 25 The Standardized Normal Table 0.9772 The Standardized The Normal table in the textbook gives the probability that z Z 0 2.00 will be less than or will The row gives the value of z to the equal to 2.00 second decimal point Z 0.00 0.01 0.02 … 0.06 The column column 0.0 shows the value 0.1 . of z to the first . 1.9. decimal point decimal 2.0 0.9750 . 9772 P(Z <= 2.00) = 0.9772, P(Z<=1.96)=0.9750 26 Find P (z ≤ 2) Find Find the area listed in the table corresponding to a z Find value of 2.00 value Starting from the top of the far left column, go down Starting to “2.0” to Read across the row z = 2.0 until under the colu...
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