Topic 6.3 The normal Distribution

Transcription

1 Topic 6.3 The normal Distribution 1

2 The Normal Curve The graph of the normal distribution depends on two factors the mean and the standard deviation. The mean of the distribution determines the location of the center of the graph, and the standard deviation determines the height and width of the graph. When the standard deviation is large, the curve is short and wide; when the standard deviation is small, the curve is tall and narrow. All normal distributions look like a symmetric, bell shaped curve, as shown below. BIGGER STANDARD DEVIATION SMALLER STANDARD DEVIATION 2

3 Properties of the normal distribution curve Probability and the Normal Curve The normal distribution is a continuous probability distribution. This has several implications for probability. The total area under the normal curve is equal to 1. The probability that a normal random variable X equals any particular value is 0. The probability that X is greater than a equals the area under the normal curve bounded by a and plus infinity (using a graphing calculator it is sufficient to assume a z score of +5). The probability that X is less than a equals the area under the normal curve bounded by a and minus infinity (using a graphing calculator it is sufficient to assume a z score of 5). 5 a a 5 3

4 Properties of the normal distribution Curve Additionally, every normal curve (regardless of its mean or standard deviation) conforms to the following "rule". About 68% of the area under the curve falls within 1 standard deviation of the mean. About 95% of the area under the curve falls within 2 standard deviations of the mean. About 99.7% of the area under the curve falls within 3 standard deviations of the mean. 4

5 Z scores A z score represents (for a particular value) it's position as the number of standard deviations above or below the mean Example: A nurse records the number of hours an infant sleeps during a day. He then records the data on a normal distribution curve shown below. The values shown on the horizontal axis differ by one standard deviation. What is the mean of the data? What is the standard deviation? What are the values for A. B, C, and D? The data value 16 is standard deviations above the mean and has a z score of The data value 13 is standard deviations the mean and has a z score of 5

6 Z score Formula Finding a Z score Where z=the number of standard deviation the score is above or below the mean. x=the score µ= the mean score σ= the standard deviation 6

7 Sample Problems 1) Tony took the three diploma exams in January and received the following results Subject Tonys Mark Provincial Average Standard Deviation Math Chem Physics Relative to the rest of the province, which test did Tony perform best on. 7

8 Sample Problems 2) In a university chemistry class the midterm produced the following results. (mean 37, standard deviation, 8) The professor decided that the marks were too low and adjusted the mean to 55, and the standard deviation to 10. If the z scores were to remain unchanged, and you received a 42 on the test, what should your new mark be after the reweighing of the test? 8

9 Sample Problems 3) A student was given a test whose scores were normally distributed. A student scored 63% which was 2.13 standard deviations above the mean. If the exam average was 57% what was the standard deviation of the test. 9

10 Sample Problems 4) The mean on a stats exam is 10k with a standard deviation of 2k 4. On the exam the student s z score was 2 his actual mark can be represented by 16k 32. What was the students actual score? 10

11 Instruction Probability and z scores The probability that an event occurs that is less than the value of a z score has been calulated in the form of a table. For example : Z= 2.15 The number represents the Area or Probability that is less than a z score of

12 Instruction There are 3 basic types of Area problems 1) Area less than a z score P(Z < 2.15) 2) Area greater than a z score P(Z > 1.06) 3) Area between two z scores P( 1.06 < Z < 2.00) 12

13 Practice Find the area under the standard normal curve for each z score interval. Give the area as a decimal and as a percent. Label the diagram. 13

14 Practice For each of the following normally distributed curves find the z score intervals which represent the area or percent given. 14

15 Problem solving with Normal Distributions EXAMPLE It was found that the mean length of 100 parts produced by a lathe was mm with a standard deviation of 0.02 mm. Find the probability that a part selected at random would have a length between mm and mm 15

16 Problem solving with Normal Distributions EXAMPLE A company pays its employees an average wage of $3.25 an hour with a standard deviation of 60 cents. If the wages are approximately normally distributed, determine 1. the proportion of the workers getting wages between $2.75 and $3.69 an hour; 2. the minimum wage of the highest 5%. 16

17 Problem solving with Normal Distributions EXAMPLE The average life of a certain type of motor is 10 years, with a standard deviation of 2 years. If the manufacturer is willing to replace only 3% of the motors that fail, how long a guarantee should he offer? Assume that the lives of the motors follow a normal distribution. 17

18 Practice A battery company produces batteries each production cycle. The mean life of each batch is 50 h and has a standard deviation of 6 h. What is the probability that a battery you purchase will last less than 36 hours? 18

19 Practice Sony CD players have a mean lifespan of 5 years before needing a repair with a standard deviation of 8 months. a) If Sony sells 4000 players this month, how many can expect repairs in the first 4 years? b) If Sony wants to repair no more than 2% of all players on warranty, how long should the warranty be? 19

20 Practice A grade 9 provincial achievement test was given and the following information was received by the schools students were above the mean. 452 students achieved an A+. Determine a) How many students wrote the test? b) If students that achieved the mean were given a mark of 50%, what was the set mark to receive an A+ if the standard deviation on the test was 12%? 20

21 Instruction Confidence intervals A confidence interval is created such that a given area is symmetrically bounded about the mean. example: a 95% confidence interval we are looking for the values so that 95% of the data fits between Z 1 and Z 2 95% 2.5% 2.5% Z 1 Z 2 21

22 Example Problem A candy company makes chocolate covered raisins. The weight of raisins per package is normally distributed with a mean of 168 g and a standard deviation of 5g. Determine the 95% confidence interval for the weight of the candy. 22

23 Practice A pop dispenser fill cans with a mean amount of 346 ml with a standard deviation of 2 ml. Determine the 95% confidence interval for the volume in each can. 23

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