Unformatted text preview: III. Probability, Random Variables and Sampling Distribution.
A. Probability (Chapter 5) B. Discrete Random Variables (Chapter 5) C. Continuous Random Variables (Chapter 6) 1. Introduction 2. Normally Distributed R.V.s 3. Standard Normal Distribution (Z) 4. Probabilities for any Normal R.V. (X) 5. Determine a Z-value given a probability. Z6. Determine an X-value given a probability. X- 5. Determine z-values given P( · )s
Case 1: Determine the z-value with P( · ) to left.
Given, P(Z Given, P(Z < z) = 0.25. What What is the z-value? zFind Find 0.2500 in the Z-table. ZThen Then z=0.675. Case 2: Determine the z-value with area α to right.
Given, Given, α = 0.25. What What is the z-value? zFind Find 0.7500 in the Z-table. ZThen Then z=0.675.
- - 5. Determine z-values given P( )s α is the area to the RIGHT of zα area RIGHT Example: Example: z0.05 Common Notation: zα -3 -2 -1 0 1 2 3 1 5. Determine z-values given P( )s Common zα values:
Complete Complete the table below. These are the most commonly used zα values.
Z0.75 Z0.50 Z0.25 Z0.20 Z0.10 Z0.05 Z0.025 Z0.01 5. Determine z-values given P( · )s
Case 3: Determine the z-values that contain area (1 - α ) between them. Case 4: Determine z-values that have areas α/2 to their left and right, respectively. - - 6. Determining x-values
Given: an area or probability. What x-value has that area to left or right? Procedure: 1. Find the area or probability in the body body of the Z-Table. 2. Determine the z-value. 3. Determine the x-value: 2 6. Determining x-values Example: A statistics professor I know is such a funny guy. He only reports exam results as zzscores. Exam 2 was distributed normally with mean of 63 and standard deviation of 20. Wh What grade did the student earn whose z-score did th zwas 1.05? 6. Determining x-values Example: An even funnier stats professor gives grades as percentiles. A student’s grade was at the 67th percentile. The mean was 81.6 and the standard deviation 10. What grade did the student earn? Z -3 0 3 A worker who earns $15 per hour is told that only 2.5% of all workers make a higher wage. If the wage is assumed to be normally distributed and the standard deviation of wage rates is $5 per hour, the average wage rates is $5 per hour, the average wage for the plant is …. (Round to one decimal place.) 3 Application: Determining crop/business losses.
Chemical contamination causes cranberry crop loss. In 2006, all harvested berries were destroyed. Damage to plants in 2006 also affected 2007 crop. How much did the grower lose? did th How can we determine loss? Application: Determining crop losses.
Cranberry production is random: variation due to weather, biennial nature of crop, disease, etc. Distribution of production – how can we measure? What parameters? – Sample – Estimate – Determine distributions
MA Cranberry Yield Distributions: Means and Standard Deviations, 2002–2007 (barrels per acre). Year Mean St. Dev. 2002 199 189 2003 311 74 2004 247 84 2005 169 93 2006 274 113 2007 220 74 Cranberry Cranberry Yield Distributions: μ and σ change each year. How can we compare the grower to other growers, on average, over time?
Yield (Bbls per acre) Compare zCompare z-scores: Bog 1: z = 0.43 Bog 2: z = 0.21 Figure 1. Average Cranberry Yields by Variety: 2001 - 2007. 250 200 150 100
Early Black Howes Stevens 50
Ben Lear 0 2001 2002 2003 2004 2005 2006 2007 4 For Bog 1, what percentile does the grower fall in? Determine the probability: P( Z < 0.43 ). Round to 2 decimal places.
1. 2. 3. 4. Illustrate. Guess. Find Z-value in the table. Determine the probability. -3 -2 -1 0 1 2 3 Dan needs four 10 foot boards, three of which must be the full 10 feet. His lumber supplier has a huge huge stock that is distributed normally with mean 121.9 inches and standard deviation of 1.5 inches. The delivery driver randomly picks 4 boards from th the stock and delivers them to Dan. What is the th Wh th probability that at least 3 of the 4 boards are at least 10 feet in length? (Use 4 decimals in your calculations. Round your final answer to one decimal place and report that digit.) 5 ...
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