8 2 Properties of a normal distribution.notebook Properties of the Normal Distribution Pages

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1 8 2 Properties of the Normal Distribution Pages normal distribution a common continuous probability distribution in which the data are distributed symmetrically and unimodally about the mean. Can be described using a single form of equation that can be used to calculate probabilities in many contexts. Examples: quality control, standardized tests, social sciences. 1

2 Example: The mode and median are equal to the mean. The smaller the standard deviation, the more the data clusters around the mean, the narrower the curve. The larger the standard deviation, the data is more spread out around the mean, the wider the curve. 2

3 For any normal distribution approximately: 68% of the data lies within of the mean. 95% of the data lies within of the mean. 99.7% of the data lies within of the mean. 3

4 Example 1 Predictions From a Normal Model Page 424 4

5 Try the following: 1. In a shipment of oranges, the oranges have diameters which are normally distributed with a mean of 8.0 cm and a standard deviation of 1.5 cm. a) Sketch a graph of a normal curve depicting this information labelling all the probabilities (%) within each standard deviation. b) Determine the probability that an orange will have a diameter: (1) less than 9.5 cm (2) between 9.5 and 11 cm (3) less than 5 cm 5

6 2. The life of a pair of Nike sneakers follows a normal distribution with a mean of 50 weeks and a standard deviation of 8 weeks. a) Sketch a normal curve depicting this situation labeling all the appropriate percentages. b) Determine the probability that a pair of Nike sneakers will wear out: (1) between 58 and 66 weeks (2) after 66 weeks (3) less than 50 weeks 6

7 standard normal distribution a normal distribution in which the mean is equal to 0 and the standard deviation is equal to 1. 7

8 z score the number of standard deviations from a data item to the mean. Used when the data in question does not fall directly on a standard deviation. Formula: Once the z score value is determined, a table is used to look up the area under the normal curve to the left of the z score which is used to determine the probability in question. 8

9 Pages

10 Determining the percent of data that lies within certain z scores: Examples: 1) to the left of z = ) to the right of z = ) between z = 2.06 and z =

11 You Try the following: Use your z score tables to determine the percent of data that lies within the given z scores: a) to the left of 2.31 b) to the left of 1.23 c) to the right of 1.88 d) to the right of 2.5 e) between 1.41 and 1.41 f) between 2.3 and

12 Calculating z scores to Determine Probabilities Examples: Example 2 Normal Probabilities Page

13 You Try: 1. A shipment of eggs has a normal distribution with a mean height of 5.20 cm and a standard deviation of 0.12 cm. Determine the probability that a randomly selected egg has a height between 5.00 cm and 5.30 cm. 13

14 2. The life of a pair of Nike sneakers follows a normal distribution with a mean of 50 weeks and a standard deviation of 8 weeks. Determine the probability that a pair of Nike sneakers lasts longer than 52 weeks. 14

15 3. In a shipment of oranges, the oranges have diameters which are normally distributed with a mean of 8.0 cm and a standard deviation of 1.5 cm. Determine the probability that an orange has a diameter less than 7 cm. 15