1) 2) 3) 4) 5) 6) 7) 8) 9) ELEMENTARY STATISTICS 3E William Navidi and Barry Monk ©McGraw-Hill Education. All rights reserved. Authorized only for instructor use in the classroom. No reproduction or further distribution permitted without the prior written consent of McGraw-Hill Education. The Standard Normal Curve Section 7.1 ©McGraw-Hill Education. Objectives 1. 2. 3. 4. Use a probability density curve to describe a population Use a normal curve to describe a normal population Find areas under the standard normal curve Find 𝑧-scores corresponding to areas under the normal curve ©McGraw-Hill Education. Objective 1 Use a probability density curve to describe a population ©McGraw-Hill Education. Probability Density Curves The following figure presents a relative frequency histogram for the particulate emissions of a sample of 65 vehicles. If we had information on the entire population, containing millions of vehicles, we could make the rectangles extremely narrow. The histogram would then look smooth and could be approximated by a curve. The curve used to describe the distribution of this variable is called the probability density curve of the random variable. The probability density curve tells what proportion of the population falls within a given interval. ©McGraw-Hill Education. Area and Probability Density Curves The area under a probability density curve between any two values 𝑎 and 𝑏 has two interpretations: • It represents the proportion of the population whose values are between 𝑎 and 𝑏. • It represents the probability that a randomly selected value from the population will be between 𝑎 and 𝑏. ©McGraw-Hill Education. Properties of Probability Density Curves The region above a single point has no width, thus no area. Therefore, if 𝑋 is a continuous random variable, 𝑃(𝑋 = 𝑎) = 0 for any number 𝑎. This means that 𝑷(𝒂 < 𝑿 < 𝒃) = 𝑷(𝒂 ≤ 𝑿 ≤ 𝒃) for any numbers 𝑎 and 𝑏. For any probability density curve, the area under the entire curve is 1, because this area represents the entire population. ©McGraw-Hill Education. Objective 2 Use a normal curve to describe a normal population ©McGraw-Hill Education. Normal Curves Probability density curves comes in many varieties, depending on the characteristics of the populations they represent. Many important statistical procedures can be carried out using only one type of probability density curve, called a normal curve. A population that is represented by a normal curve is said to be normally distributed, or to have a normal distribution. ©McGraw-Hill Education. Properties of Normal Curves The population mean determines the location of the peak. The population standard deviation measures the spread of the population. Therefore, the normal curve is wide and flat when the population standard deviation is large, and tall and narrow when the population standard deviation is small. The mean and median of a normal distribution are both equal to the mode. ©McGraw-Hill Education. Empirical Rule The normal distribution follows the Empirical Rule. ©McGraw-Hill Education. Objective 3 Find areas under the standard normal curve *(TI-84 PLUS) ©McGraw-Hill Education. Standard Normal Curve A normal distribution can have any mean and any positive standard deviation. However, the normal distribution with a mean of 0 and standard deviation of 1 is known as the standard normal distribution. ©McGraw-Hill Education. 𝑍-Scores When finding an area under the standard normal curve, we use the letter 𝑧 to indicate a value on the horizontal axis beneath the curve. We refer to such a value as a 𝒛-score. Since the mean of the standard normal distribution is 0: • The mean has a 𝒛-score of 0. • Points on the horizontal axis to the left of the mean have negative 𝒛scores. • Points to the right of the mean have positive 𝒛-scores. ©McGraw-Hill Education. Finding Areas with the TI-84 PLUS On the TI-84 PLUS calculator, the normalcdf command is used to find areas under a normal curve. Four numbers must be used as the input. The first entry is the lower bound of the area. The second entry is the upper bound of the area. The last two entries are the mean and standard deviation. This command is accessed by pressing 2nd, Vars. ©McGraw-Hill Education. Ex 1: Area Under Standard Normal (TI-84) Find the area to the left of 𝑧 = 1.26. Note the there is no lower endpoint, therefore we use -1E99 (or -10^99) which represents negative 1 followed by 99 zeroes. We select the normalcdf command and enter -1E99 as the lower endpoint, 1.26 as the upper endpoint, 0 as the mean and 1 as the standard deviation. The area to the left of 𝑧 = 1.26 is 0.8962. ©McGraw-Hill Education. Ex 2: Area Under Standard Normal (TI-84) Find the area to the right of 𝑧 = –0.58. Note the there is no upper endpoint, therefore we use 1E99 (or 10^99) which represents the large number 1 followed by 99 zeroes. We select the normalcdf command and enter –0.58 as the lower endpoint, 1E99 as the upper endpoint, 0 as the mean and 1 as the standard deviation. The area to the right of 𝑧 = –0.58 is 0.7190. ©McGraw-Hill Education. Ex 3: Area Under Standard Normal (TI-84) Find the area between 𝑧 = –1.45 and 𝑧 = 0.42. We select the normalcdf command and enter –1.45 as the lower endpoint, 0.42 as the upper endpoint, 0 as the mean and 1 as the standard deviation. The area between 𝑧 = –1.45 and 𝑧 = 0.42 is 0.5892. ©McGraw-Hill Education. Objective 4 Find 𝑧-scores corresponding to areas under the normal curve *(TI-84 PLUS) ©McGraw-Hill Education. 𝑍-scores (Normal Values) From Areas We have been finding areas under the normal curve from given 𝑧-scores. Many problems require us to go in the reverse direction. That is, if we are given an area, we need to find the 𝑧-score (value from the population ) that corresponds to that area under the standard normal curve. ©McGraw-Hill Education. Normal Values From Areas on the TI-84 PLUS The invNorm command on the TI-84 PLUS calculator returns the value from the normal population with a given area to its left. This command takes three values as its input. The first value is the area to the left, the second and third values are the mean and standard deviation, respectively. This command is accessed by pressing 2nd, Vars. ©McGraw-Hill Education. Example 1: Finding 𝑧-Scores (TI-84 PLUS) Find the 𝑧-score that has an area of 0.26 to its left. We select the invNorm command and enter 0.26 as the area to the left, 0 as the mean, and 1 as the standard deviation. The z-score with an area of 0.26 to its left is approximately –0.64. ©McGraw-Hill Education. Example 2: Finding 𝑧-Scores (TI-84 PLUS) Find the 𝑧-score that has an area of 0.68 to its right. Since the area to the right is 0.68, the area to the left is 1 – 0.68 = 0.32. We use the invNorm command with 0.32 as the area to the left, 0 as the mean, and 1 as the standard deviation. The z-score with an area of 0.68 to its right is approximately –0.47. ©McGraw-Hill Education. Example 3: Finding 𝑧-Scores (TI-84 PLUS) Find the 𝑧-scores that bound the middle 95% of the area under the standard normal curve. We first sketch a normal curve and shade in the given area. Label the 𝑧-score on the left 𝑧1 and the 𝑧-score on the right 𝑧2. Now, since the area in the middle is 0.95, the area in the two tails combined is 0.05. Half of that area, which is 0.025, is to the left of 𝑧1. We use the invNorm command with 0.025 as the area to the left, 0 as the mean, and 1 as the standard deviation. We find that 𝑧1 = –1.96. By symmetry, 𝑧2 = 1.96. ©McGraw-Hill Education. You Should Know . . . • How to use a probability density curve to describe a population • How the shape of a normal curve is affected by the mean and standard deviation • How to find areas under a normal curve (Calculator) • How to find 𝑧-scores corresponding to areas under a normal curve (Calculator) ©McGraw-Hill Education.
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