8.2 The Normal Probability Distribution• Properties of a Normal Distribution

Every normal distribution has a mean, μ and a standard deviation, δ. The graph (bell-shaped curve) is symmetrical about the mean. The horizontal axis is an asymptote.The total area under the curve is 1.

• The 68-95-99 Rule• A(z) = area under the standard normal curve on the left side of a vertical line corresponding to z.

• z =

Example 1 The answers are: -1.09; -1.04; -1.48; 1.23Find the value of a 1) P(z < a ) = 0.1379 2) P(z > a ) = 0.85083) P(a < z < 0 ) = 0.4306 4) P(0 < z < a ) = 0.3907

Example 2 The answer is BIf Z has a standard normal distribution with P(a Z b ) = 0.1, where -3 a b 3, which of the following statements must be true?A. P(Z a ) > 0.1 B. P(Z > a ) > 0.1 C. P(Z b ) 0.1 D. P(Z > a ) 0.1

Example 3 The answer is DThe shaded area in the diagram represents 23.97% of the area under the curve. The value of z1

isA. -0.92B. -0.71C. -0.64D. -0.52

-1.54 z1 0 z

Example 4A manufacturer produces batteries with a mean life of 1500 hours and a standard deviation of 100 hours. Determine a 99% confidence interval for the life of a battery. z1 z2

Solution: 99% confidence interval means the shaded area is 99%, so the area left of z1 is 0.5% and also the area right of z2 is 0.5%. z1 = -2.58 and z2 = 2.58

Because z = , so x = z + μ.

Substituting the z with z1 and z2, we get xmin = 1242 h and xmax = 1758 h.So the life of a battery of 99% confidence interval is between 1242 h and 1758

h.

Example 5The weight of a large shipment of coconuts is normally distributed with a standard deviation of 0.71kg. the probability that a coconut weight less than 2.7kg is 10.2%. The mean of the shipment is_____?

Solution: from the information----a coconut weight less than 2.7kg is 10.2%,

we get z1 = -1.27. Because z1 = = -1.27, so μ = 3.60

Example 6 The answer is CThe results of a district-wide science test were normally distributed with a mean of 80 and a standard deviation of 16. Determine the percent of scores that would be expected to be between 70 and 90.A. 27% B. 40% C. 47% D. 73%

Example 7 The answer is BThe results for a test are normally distributed. The mean score is 67 with a standard deviation of 9. If the top 10% of the students receive an “A”, what is the minimum mark needed to receive an “A”?A. 55 B. 79 C. 86 D. 90

Example 8 The answer is AThe mark distributions of Biology and Chemistry students are given by the two normal curves below.

Chemistry

= 63% = 69% = 18% = 5%

John has the same z-score standing in Biology and Chemistry. If his mark in Biology is 74%, what is his mark in Chemistry?A. 72% B. 73% C. 74% D. 75%