5-Minute Check on Section 2-1a Click the mouse button or press the Space Bar to display the answers. 1.What does a z-score represent? 2.According to Chebyshev’s Inequality, how much of the data must lie within two standard deviations of any distribution? Given the following z-scores on the test: Charlie, 1.23; Tommy, -1.62; and Amy, Who had the better test score? 4.Who was farthest away from the mean? 5.What does the IQR represent? 6.What percentile is the person who is ranked 12 th in a class of 98? number of standard deviations away from the mean 1 – 1/2 2 = 1 – 0.25 = 0.75 so 75% of the data Charlie; his was the highest z-score Tommy ; his was the highest |z-score| The width of the middle 50% of the data; a single number 1 – 12/98 = 1 – = or 88%-tile

Density Curve In Chapter 1, you learned how to plot a dataset to describe its shape, center, spread, etc Sometimes, the overall pattern of a large number of observations is so regular that we can describe it using a smooth curve Density Curve: An idealized description of the overall pattern of a distribution. Area underneath = 1, representing 100% of observations.

Density Curves Density Curves come in many different shapes; symmetric, skewed, uniform, etc The area of a region of a density curve represents the % of observations that fall in that region The median of a density curve cuts the area in half The mean of a density curve is its “balance point”

Describing a Density Curve To describe a density curve focus on: Shape –Skewed (right or left – direction toward the tail) –Symmetric (mound-shaped or uniform) Unusual Characteristics –Bi-modal, outliers Center –Mean (symmetric) or median (skewed) Spread –Standard deviation, IQR, or range

Mean, Median, Mode In the following graphs which letter represents the mean, the median and the mode? Describe the distributions

Mean, Median, Mode (a) A: mode, B: median, C: mean Distribution is slightly skewed right (b) A: mean, median and mode (B and C – nothing) Distribution is symmetric (mound shaped) (c) A: mean, B: median, C: mode Distribution is very skewed left

Uniform PDF ●Sometimes we want to model a random variable that is equally likely between two limits ●When “every number” is equally likely in an interval, this is a uniform probability distribution –Any specific number has a zero probability of occurring –The mathematically correct way to phrase this is that any two intervals of equal length have the same probability ●Examples  Choose a random time … the number of seconds past the minute is random number in the interval from 0 to 60  Observe a tire rolling at a high rate of speed … choose a random time … the angle of the tire valve to the vertical is a random number in the interval from 0 to 360

Uniform Distribution All values have an equal likelihood of occurring Common examples: 6-sided die or a coin This is an example of random numbers between 0 and 1 This is a function on your calculator Note that the area under the curve is still 1

Continuous Uniform PDF Discrete Uniform PDF P(x=0) = 0.25 P(x=1) = 0.25 P(x=2) = 0.25 P(x=3) = 0.25 P(x=1) = 0 P(x ≤ 1) = 0.33 P(x ≤ 2) = 0.66 P(x ≤ 3) = 1.00

Example 1 A random number generator on calculators randomly generates a number between 0 and 1. The random variable X, the number generated, follows a uniform distribution a.Draw a graph of this distribution b.What is the percentage (0<X<0.2)? c.What is the percentage (0.25<X<0.6)? d.What is the percentage > 0.95? e.Use calculator to generate 200 random numbers Math  prb  rand(200) STO L3 then 1varStat L3 1 1

Statistics and Parameters Parameters are of Populations –Population mean is μ –Population standard deviation is σ Statistics are of Samples –Sample mean is called x-bar or x –Sample standard deviation is s

Summary and Homework Summary –We can describe the overall pattern of a distribution using a density curve –The area under any density curve = 1. This represents 100% of observations –Areas on a density curve represent % of observations over certain regions –Median divides area under curve in half –Mean is the “balance point” of the curve –Skewness draws the mean toward the tail Homework –Day 2: pg probs 2-9, 10, 12, 13, pg probs 15, 18