Transcript 2.1 part 2
Finish Section 2.1 Density Curves 1. Always plot your data: dotplot, stemplot, histogram… 2. Look for overall pattern (SOCS) 3. Calculate numerical summary to describe Add a step… 4. Sometimes the overall pattern is so regular we can describe it by a smooth curve. A Density Curve is a curve that: • Is always on or above the horizontal axis • Has an area of 1 underneath it Things to note… • The mean is the physical balance point of a density curve or a histogram • The median is where the areas on both sides of it are equal Don’t forget… • The mean and the median are equal for symmetric density curves! Start Section 2.2 Norman Curves! Normal Distributions = Normal Curves • Any particular Normal distribution is completely specified by two numbers: its mean (𝜇) and standard deviation (𝜎). • We abbreviate the Normal distribution with mean 𝜇 and standard deviation 𝜎 as N(𝝁, 𝝈) The 68-95-99.7 Rule Other images to explain the same thing…in case it helps! Example • A pair of running shoes lasts on average 450 miles, with a standard deviation of 50 miles. Use the 68-95-99.7 rule to find the probability that a new pair of running shoes will have the following lifespans. Between 400-500 miles More than 550 miles Example • A survey of 1000 U.S. gas stations found that the price charged for a gallon of regular gas can be closely approximated by a normal distribution with a mean of $1.90 and a standard deviation of $0.20. How many of the stations charge: a. between $1.50 and $2.30 for a gallon of regular gas? b. less than $2.10 for a gallon of regular gas? c. more than $2.30 for a gallon of regular gas? Example • A vegetable distributor knows that during the month of August, the weights of its tomatoes were normally distributed with a mean of 0.61 pound and a standard deviation of 0.15 pound. a. What percent of the tomatoes weighed less than 0.76 pound? b. In a shipment of 6000 tomatoes, how many tomatoes can be expected to weigh more than 0.31 pound? c. In a shipment of 4500 tomatoes, how many tomatoes can be expected to weigh between 0.31 and 0.91 pound? How does this relate to z-scores? • If the distribution happens to be normal we can find the area under the curve and percentiles by using z-scores! • Standard Normal distribution – mean of 0 and standard deviation of 1. •𝒛 = 𝒙−𝝁 𝝈 Table A in your book… • The table entry for each z-score is the area under the curve to the left of z. • If we wanted the area to the right, we would have to subtract from 1 or 100% Examples • Find the proportion of observations that are less than 0.81 • Find the proportion of observations that are greater than -1.78 • Find the proportion of observations that are less than 2.005 • Find the proportion of observations that are greater than 1.53 • Find the proportion of observations that are between -1.25 and 0.81 Homework • Pg 107 (19-24, 27, 28)