THINK ABOUT IT!!!!!!! If a satellite crashes at a random point on earth, what is the probability it will crash on land, if there are 54,225,000 square miles of land and 142,715,000 square miles of water?

Section 6.1 Review and Preview

Preview Chapter focus is on: Continuous random variables Normal distributions Standard Normal Distributions

The Standard Normal Distribution Section 6.2 The Standard Normal Distribution

Important Notes The standard normal distribution has three properties: Its graph is bell-shaped. Its mean is equal to 0 ( = 0). Its standard deviation is equal to 1 ( = 1). Objectives: Develop the skill to find areas (or probabilities or relative frequencies) corresponding to various regions under the graph of the standard normal distribution. Find z-scores that correspond to area under the graph.

Uniform Distribution A continuous random variable has a uniform distribution if its values are spread evenly over the range of probabilities. The graph of a uniform distribution results in a rectangular shape, and has an area of 1 square unit.

Example 1: The Newport Power and Light Company provides electricity with voltage levels that are uniformly distributed between 123.0 volts and 125.0 volts. That is, any voltage amount between 123.0 volts and 125.0 volts is possible, and all of the possible values are equally likely. If we randomly select one of the voltage levels and represent its value by the random variable x, then x has a distribution that can be graphed as in Figure 6.2 below. Assume a voltage level between 123.0 volts and 125.0 volts is randomly selected, find the probability that the given voltage level is selected. Greater that 123.3 volts. b) Less than 124.2 volts. c) Between 123.7 volts and 124.6 volts. x Area = 1 125.0 123.0 P(x) 0.5 Voltage

Density Curve A density curve is the graph of a continuous probability distribution. It must satisfy the following properties: The total area under the curve must equal 1. Every point on the curve must have a vertical height that is 0 or greater. (That is, the curve cannot fall below the x-axis.) Because the total area under the density curve is equal to 1, there is a correspondence between area and probability.

Warm Up Use the following information for numbers 1 – 6: A random variable X can take values from 670 to 780 and has a uniform distribution.  1. What is the height of the distribution?   3. What is the probability that X is greater than 704? 5. What is the probability that X is less than 704? 2. What is the probability that X is between 704 and 727?   4. What is the probability that X is greater than 780? 6. What is the probability that X is less than 670?

Standard Normal Distribution The standard normal distribution is a normal probability distribution with  = 0 and  = 1. The total area under its density curve is equal to 1.

Method for Finding Normal Distribution Areas (When being asked to find an area or probability you will use this same method) , mean, std. dev)

Notation P(a < z < b) denotes the probability that the z score is between a and b. To find this probability in your calculator, type: normalcdf(a, b, 0, 1) P(z > a) denotes the probability that the z score is greater than a. To find this probability in your calculator, type: normalcdf(a, 99999, 0, 1) P(z < a) denotes the probability that the z score is less than a. To find this probability in your calculator, type: normalcdf(–99999, a, 0, 1) a   b a   a  

Example 2: Assume that the readings on the thermometers are normally distributed with a mean of 0˚C and a standard deviation of 1.00˚C. Find the indicated probability, where z is the reading in degrees. a) P(–1.96 < z < 1.96) b) P(–2.575 < z < 2.575) c) P(z > 1.32)

Example 3: Find the area of the shaded region Example 3: Find the area of the shaded region. The graph depicts the standard normal distribution with mean 0 and standard deviation 1. z=1.13

Example 4: Find the area of the shaded region Example 4: Find the area of the shaded region. The graph depicts the standard normal distribution with mean 0 and standard deviation 1. z = –1.88 z = 1.88

Example 5: The Precision Scientific Instrument Company manufactures thermometers that are supposed to give readings of 0°C at the freezing point of water. Tests on a large sample of these instruments reveal that at the freezing point of water, some thermometers give readings below 0° (denoted by negative numbers) and some give readings above 0° (denoted by positive numbers). Assume that the mean reading is 0° C and the standard deviation of the readings is 1.00°C. Also assume that the readings are normally distributed. If one thermometer is randomly selected, find the probability that, at the freezing point of water, the reading is…….. less than 1.27°. greater than –1.23° c) between -2.00 ° and 1.50 °

When Given an Area or Probability Finding z Scores When Given an Area or Probability Methods for Finding z scores when given Areas or Probabilities   1. Sketch a normal distribution curve, enter the given probability or percentage in the appropriate region of the graph, and identify the z score(s) being sought. 2. Determine the area to the left and enter the following into your calculator: invNorm(area to the left, 0, 1) *2nd, vars, invnorm (#3) 3. Refer to the sketch of the curve to verify that the solution makes sense in the context of the graph and the context of the problem.

Finding z Scores When Given an Area or Probability (z score will be positive) 5% or 0.05 Find the 95th Percentile by using the InvNorm feature on the TI 81/84 Calculator using area to the left. Type: InvNorm(0.95,0,1) to get the positive z-score.

(One z score will be negative and the other positive) Finding the Bottom 2.5% and Upper 2.5%, again using the InvNorm feature on the TI 83/84 Calculator using area to the left. Type InvNorm(0.025,0,1) to get the negative z-score.

Example 6: Find the indicated z score Example 6: Find the indicated z score. The graph depicts the standard normal distribution with mean 0 and standard deviation 1. a) The shaded area is 0.0401. b) The shaded area is 0.4483.

Critical Values: For a normal distribution, a critical value is a z score on the borderline separating the z scores that are likely to occur from those that are unlikely. Common critical values are z = –1.96 and z = 1.96 and they are obtained as shown on a previous slide.

Notation The expression zα denotes the z score with an area of α to its right. In a normal curve, the α (alpha) represents the area to the right of the z score that is to be found. To find this z score in your calculator, type: invNorm(1 – α, 0, 1)

Example 7: Find the indicated value. a) z0.21 b) z0.03