Normal Curves & Sampling Distributions

Transcription

1 Chapter 7 Name Normal Curves & Sampling Distributions Section 7.1 Graphs of Normal Probability Distributions Objective: In this lesson you learned how to graph a normal curve and apply the empirical rule to solve real-world problems. Important Vocabulary Normal Curve Empirical Rule What is the graph of a normal distribution called? What is another name for a normal curve? Focus Points: Graph a normal curve and summarize its important properties Apply the empirical rule to solve realworld problems What does the parameter σ (standard deviation) control? If σ is large what happens to the curve? If σ is small? Important Properties of a Normal Curve

2 Why is the area under a normal curve important? Empirical Rule Figure 7-1 How does the application of the empirical rule vary from Chebyshev s Theorem? What does a Normal Distribution tell us? 2

3 Section 7.1 Example Graphs of Normal Probability Distributions ( 1 ) The yearly wheat yield per acre on a particular farm is normally distributed with mean μ = 35 bushels and standard deviation σ = 8 bushels. a. Shade the area under the curve, in the figure to the right, that represents the probability that an acre will yield between 19 and 35 bushels. b. Is the area the same as the area between μ 2σ and μ? c. Use Figure 7-1 (from your notes) to find the percentage of area over the interval between 19 and 35. d. What is the probability that the yield will be between 19 and 35 bushels per acre? 3

4 Section 7.2 Standard Units and Areas Under the Standard Normal Distribution Objective: In this lesson you learned to convert between raw scores and z-scores; graph standard normal distributions; and find areas under the standard normal curve. Important Vocabulary z-score Standard Score Raw Score Standard Normal Distribution I. z-scores and Raw Scores In what ways do normal distributions vary from each other? Focus Points: Given μ and σ, convert raw scores to z-scores Given μ and σ, convert z-scores to raw scores The z-value In standard units, what is the mean of the original distribution? A x-value that is above the original mean is what? A x-value that is below the original mean is what? What is a raw score? How is it calculated? II. Standard Normal Distribution The standard normal distribution Focus Point: Graph the standard normal distribution 4

5 What does the Standard Normal Distribution tell us? III. Areas Under the Standard Normal Curve What is the advantage to converting a normal distribution to a standard distribution? Focus Point: Find areas under the Standard Normal Curve Why are the area(s) under the curve important? IV. Using a Standard Normal Distribution Table How to use a Left-Tail style Standard Normal Distribution Table

6 Section 7.2 Examples Standard Units and Areas Under the Standard Normal Distribution ( 1 ) Rod figures that it takes an average (mean) of 17 minutes with a standard deviation of 3 minutes to drive from home, park the car, and walk to an early morning class. a. One day it took Rod 21 minutes to get to class. How many standard deviations from the average is that? Is the z-value positive or negative? Explain why the z-value should be either positive or negative. b. What commuting time corresponds to a standard score of z = 2.5? Could Rod count on making it to class in this amount of time or less? ( 2 ) Table 3, Areas of a Standard Normal Distribution, is located in the Appendix as well as in the endpapers of the text. Spend a little time studying the table, and then answer these questions. a. As z-values increase, do the areas to the left of z increase? b. If a z value is negative, is the area to the left of z less than ? c. If a z value is positive, is the area to the left of z greater than ? ( 3 ) Let z be a random variable with a standard normal distribution. a. P(z 1.15) refers to the probability that z-values lie to the right of Draw and shade the corresponding area under the standard normal curve and find P(z 1.15). b. Find P( 1.78 z 0.35). First, sketch the area under the standard normal curve corresponding to P( 1.78 z 0.35). 6

7 Section 7.3 Areas Under Any Normal Curve Objective: In this lesson you learned to compute probabilities of standardized events and find a z-score from a given normal probability. I. Normal Distribution Areas How to work with Normal Distributions Focus Point: Compute the probability of standardized events II. Inverse Normal Distribution What is inverse normal probability distribution? Focus Point: Find a z-score from a given normal probability (inverse normal) III. Checking for Normality 1. Histogram: 2. Outliers: 3. Skewness: 7

8 Section 7.3 Examples Areas Under Any Normal Curve ( 1 ) The life span of a rechargeable battery is the time before the battery must be replaced because it no longer holds a charge. One tablet computer model has a battery with a life span that is normally distributed with a mean of 2.3 years and a standard deviation of 0.4 year. What is the probability that the battery will have to be replaced during the guarantee period of 2 years? a. Let x represent the battery life span. The statement that the battery needs to be replaced during the 2-year guarantee period means the life span is less than 2 years, or x 2. Convert this statement to a statement about z. b. Indicate the area to be found, in the figure below, by shading. Does this area correspond to the probability that z 0.75? c. Use the Standard Normal Distribution table to find P(z 0.75). d. What is the probability that the battery will fail before the end of the guarantee period? ( 2 ) Find the z-value such that 3% of the area under the standard normal curve lies to the right of z. a. Draw a sketch of the standard normal distribution showing the described area. b. Find the area to the left of z. c. Look up the area in the Standard Normal Distribution table and find the corresponding z-value. 8

9 d. Suppose the time to complete a test is normally distributed with μ = 40 minutes and σ = 5 minutes. After how many minutes can we expect all but about 3% of the tests to be completed? e. Use the Standard Normal Distribution table to find a z-value such that 3% of the area under the standard normal curve lies to the left of z. 9

10 Section 7.4 Sampling Distributions Objective: In this lesson you learned to construct a relative frequency distribution for x values from raw data. Important Vocabulary Statistic Parameter Sampling Distribution What is a population, in terms of statistics? What is a sample? What is a statistic? What is a parameter? Focus Points: Review such commonly used terms as random sample, relative frequency, parameter, statistic, and sampling distribution From raw data, construct a relative frequency distribution for x values and compare the result to a theoretical sampling distribution 10

11 Types of Inferences 1. Estimation: 2. Testing: 3. Regression: A sampling distribution What does a sampling distribution tell us? 11

12 Section 7.4 Examples Sampling Distributions ( 1 ) What is a population parameter? Give an example. ( 2 ) What is a sample statistic? Give an example. ( 3 ) What is a sampling distribution? 12

13 Section 7.5 The Central Limit Theorem Objective: In this lesson you learned to construct theoretical sampling distributions for the statistic x ; use sample estimates to construct a good approximate sampling distribution for the statistic x ; the underlying meaning of the central limit theorem. Important Vocabulary Standard Error Central Limit Theorem Large Sample I. The x Distribution, Given x is Normal Theorem 7.1 For a Normal Probability Distribution Focus Point: For a normal distribution, use μ and σ to construct the theoretical sampling distribution for the statistic x What is the standard deviation of a statistic referred to as? The standard error 13

14 II. The x Distribution, Given x Has an Distribution Theorem 7.2 The Central Limit Theorem Focus Point: For large samples, use sample estimates to construct a good approximate sampling distribution for the statistic x How large should a sample be for the x distribution to appear normal? What does the Central Limit Theorem tell us? Using the Central Limit Theorem to convert the x Distribution 14

15 How to find Probabilities Regarding x

16 Section 7.5 Examples The Central Limit Theorem ( 1 ) Using the Central Limit Theorem a. Suppose x has a normal distribution with mean μ = 18 and standard deviation σ = 13. If you draw random samples of size 5 from the x distribution and x represents the sample mean, what can you say about the x distribution? How could you standardize the x distribution? b. Suppose you know that the x distribution has mean μ = 75 and standard deviation σ = 12, but you have no information as to whether or not the x distribution is normal. If you draw samples of size 30 from the x distribution and x represents the sample mean, what can you say about the x distribution? How could you standardize the x distribution? c. Suppose you did not know that x had a normal distribution. Would you be justified in saying that the x distribution is approximately normal if the sample size were n = 8? 16

17 ( 2 ) In mountain country, major highways sometimes use tunnels instead of long, winding roads over high passes. However, too many vehicles in a tunnel at a time can cause a hazardous situation. Traffic engineers are studying a long tunnel in Colorado. If x represents the time for a vehicle to go through the tunnel, it is known that the x distribution has a mean μ = 12.1 minutes and standard deviation σ = 3.8 minutes under ordinary traffic conditions. From a histogram of x values, it was found that the x distribution is mound-shaped with some symmetry about the mean. Engineers have calculated that, on average, vehicle should spend from 11 to 13 minutes in the tunnel. If the time is less than 11 minutes, traffic is moving too fast for safe travel in the tunnel. If the time is more than 13 minutes, there is a problem of bad air quality (too much carbon monoxide and other pollutants). Under ordinary conditions, there are about 50 vehicles in the tunnel at one time. What is the probability that the mean time for 50 vehicles in the tunnel will be from 11 to 13 minutes? a. Let x represent the sample mean based on samples of size 50. Describe the x distribution. b. Find P(11 < x < 13). c. Interpret your answer to part (b). 17

18 Section 7.6 Normal Approximation to Binomial Distribution and to p Distribution Objective: In this lesson you learned how to compute μ and σ for the normal approximation; use continuity correction to convert a range of r values to a corresponding range of normal x values; convert the x values to a range of standardized z scores and find desired probabilities; and describe the sampling distribution for proportions p. I. Normal Approximations Normal Approximation to the Binomial Distribution Consider a binomial distribution where How to make a Continuity Correction Focus Point: State assumptions needed to use the normal approximation to the binomial distribution Compute μ and σ for the normal approximation Use the continuity correction to convert a range of r values to normal x values, then into standardized z scores to find desired probabilities What does a Normal Approximation to the Binomial tell us? 18

19 II. Sampling Distributions for Proportions Sampling Distribution for the Proportion p = r n Focus Point: Describe the sampling distribution for proportions p What does a p Sampling Distribution tell us? 19

20 Section 7.6 Examples Normal Approximation to Binomial Distribution and to p Distribution ( 1 ) From many years of observation, a biologist knows that the probability is only 0.65 than any given Arctic tern (a type of bird) will survive the migration from its summer nesting area to its winter feeding grounds. A random sample of 500 Arctic terns were banded at their summer nesting area. Use the normal approximation to the binomial and the following steps to find the probability that between 310 and 340 of the banded Arctic terns will survive the migration. Let r be the number of surviving terns. a. To approximate P(310 r 340), we use the normal curve with μ = and σ =. b. P(310 r 340) is approximately equal to P( x ), where x is a variable from the normal distribution described in part (a). c. Convert the condition x to a condition in standard units. d. P(310 r 340) = P(309.5 x 340.5) = P( 1.45 z 1.45) =. e. Will the normal distribution make a good approximation to the binomial for the problem? Explain your answer. 20