University:
High School
Course:
High School Statistics
Academic year:

2022
Views:

242

Pages:

4
Author:

Diamond Bolton

11.1 Using Normal Distributions A. VOCABULARY & CONCEPTS 1. Standard Deviation: A measure of how much a typical value in a data set differs from the mean. σ = √((X₁-µ)² + (X₂-µ)² + ... + (Xn-µ)²)/n ↑ Standard deviation "Sigma" ↑ mean "mu" ↑ of values in data set 2. Normal Distribution: Bell-shaped curve, symmetric about the mean A diagram of a bell curve with the following labels: μ (mean) μ ± σ (mean plus or minus one standard deviation) μ ± 2σ (mean plus or minus two standard deviations) μ ± 3σ (mean plus or minus three standard deviations) Percentages of data within each standard deviation: 68%, 95%, 99.7% mean = median 3. Skewed Left: Description: A bell-shaped curve that is not symmetrical. The tail on the left side is longer than the tail on the right side. Characteristics: Not a normal distribution Mean < Median 4. Skewed Right: Description: A bell-shaped curve that is not symmetrical. The tail on the right side is longer than the tail on the left side. Characteristics: Not a normal distribution Mean > Median B. EXAMPLES 1. Finding Normal Probability a) P(μ - 2σ ≤ x ≤ μ) This means: What is the probability that a randomly selected x-value lies between μ - 2σ and μ? Diagram: A bell-shaped curve with μ (mean) and μ - 2σ (mean minus 2 standard deviations) labeled. The area under the curve between these two points is shaded. Calculation: P(μ - 2σ ≤ x ≤ μ) = 13.5% + 34% = 47.5% b) P(x ≤ μ + 2σ) This means: What is the probability that a randomly selected x-value is less than or equal to the mean + 2 standard deviations. Diagram: A bell-shaped curve with μ (mean) and μ + 2σ (mean plus 2 standard deviations) labeled. The area under the curve to the left of μ + 2σ is shaded. Calculation: 95% + 2.35% + 0.15% = 97.5% P = 0.975 c) P(μ - σ ≤ x ≤ μ + 3σ) Diagram: A bell-shaped curve with μ (mean), μ - σ (mean minus one standard deviation), and μ + 3σ (mean plus three standard deviations) labeled. The area under the curve between μ - σ and μ + 3σ is shaded. Calculation: 68% + 13.5% + 2.35% = 83.85% P = 0.8385 2. Interpreting Normally Distributed Data Context: The scores for a state's peace officer standards and training test are normally distributed with a mean of 55 and a standard deviation of 12. The test scores range from 0 to 100. a) About what percent of the people taking the test have scores between 43 and 67? Diagram: A bell-shaped curve with μ = 55, σ = 12, and the range 43 to 67 marked on the x-axis. Calculation: 68% (This range represents one standard deviation below and above the mean) Answer: Approximately 68% of the people taking the test have scores between 43 and 67.