Lecture 6: Chapter 6 Summary

Transcription

1 1 Lecture 6: Chapter 6 Summary Z-score: Is the distance of each data value from the mean in standard deviation Standardizes data values Standardization changes the mean and the standard deviation: o Z has mean 0 and standard deviation of 1. The farther the data value is from the mean, the more unusual it might me. For example recall our donuts calories data: We had one observation with 400 calories. Let s find out the Z-score for the 400 calories. Recall the mean = 251 and S = = 2.98

2 2 Comparing the boxplot with normal density curve: Why 1.5 x IQR? The values that are beyond the inner fences: Q1 1.5(IQR) and Q (IQR) Correspond to z-scores beyond and and are considered to be a potential outlier (these are z-scores that correspond to extreme values).

3 3 Normal model and Density curve: To model the frequency distribution of a quantitative variable, use histogram We can imagine a curve going through the rectangles Better to look at relative frequencies (proportions) Add up all the relative frequencies (weight of each rectangle) and get total area for the rectangles. This total is Density curve: o Describes the overall pattern of a distribution o Is above the horizontal axis o Has area exactly 1 underneath it (as explained above by adding the area of each rectangle); covering 100% of the data observations o Eg. Normal density curve. o The curve below shows the density curve for calories in Tim Horton s donuts. The area of the shaded region is the proportion of donuts with calories between 270 and 340.

4 4 Normal Distribution: Symmetric, unimodal, bell-shaped curves, are known as normal curves that describe normal distributions. The density (shape) for a normal distribution has mean mu) and standard deviation (pronounced sigma). and are Greek letters and denote population parameters. (pronounced and S are statistics and denote sample mean and sample standard deviations respectively. The mean is located in the centre of the symmetric, unimodal, bell-shaped curve and is the same as the median (and the mode). Suppose X is an observation from a normal distribution with and. We write X N(, ) We can standardize this X and express it as Z. Z N(0, 1) Z with mean 0 and standard deviation of 1. When we know X is normal, we convert the X to Z and use the Z table to find an area of interest (See examples coming up). Z table is posted on the main site: See on the left side info: the tables you ll get on the exam Open the file and for this chapter we use the Z Table (Normal distribution) The area for this table is to the left The first column from the left and the first row (headings) joint together make a z-score For example, join -3.8, the first value in the first column, from the left, with 0.09, the value in the first row (headings): You have got a Z score of and the area below this value is Note that there is no weight, area, or mass function at a particular value. So, we always look at an area above or below the Z-score. The graph is symmetric (Z distribution has mean zero; half the values are above the mean and the other half are below the mean). This means that area above Z-score of for example, 3.89, is the same as the area below the Z-score of -3.89

5 5 Example: Suppose calories of Tim Horton s donuts have a normal distribution with mean 251 and standard deviation a) what proportion of the donuts have calories less than 230? We ll do this on the board a) what proportion of the donuts have calories between 270 and 340 calories?

6 6 c) Asal would like to buy a donut from the low 10 th percentile calories. What is the most calories of a donut in the low 10 th percentile? REVERSE LOOK UP IN THE Z TABLE

7 7 Normal Quantile Plots Use histogram, stem-and-leaf display, or boxplot to check to see if the distribution is about symmetric and unimodal. We use normal quantile or normal probability plot to better determine the adequacy of a normal model. If points lie close to the straight line, the plot indicates that the data are normal. Outliers will be plotted individually away from the rest of the data. Let s visit the donuts calories distribution. Boxplot: Histogram: Stem-and-leaf display: Leaf unit = 10 1 : : : : : 4 : 0

8 8 Normal Quantile Plot: StatCrunch> Graph> QQ plot> Select column(s): Calories click compute * notice the value 400 calories is plotted away from the rest of the data * all other values are close to the straight line

9 9 The % Rule (Empirical Rule) The normal distribution with the mean x-bar and the standard deviation S: About (approximately) 68% of the observations in the data are within 1 standard deviation from the mean: ( About (approximately) 95% of the observations in the data are within 2 standard deviation from the mean: ( About (approximately) 99.7% or almost all of the observations in the data are within 3 standard deviation from the mean: ( Note: If the empirical rule hold (all conditions above are met), it does not mean the distribution is mound shape. If the distribution is mound shape, the empirical rule hold

10 10 Recall donut calories example with mean 251 and standard deviation of 50.06: Calories ~ 68% of 30 observation is: 0.68 x 30 = 20.4 ( 20 to 21 observation): ( = ( , ) = (200.94, ) So, given this interval, we include calories of donuts from 210 to 300. There are 21 donuts in this interval. ~ 95% of 30 observation is: 0.95 x 30 = 28.5 ( 28 to 29 observation): ( = (251 (2*50.06), (2*50.06)) = (150.88, ) So, given this interval, we include calories of donuts from 180 to 340. There are 29 donuts in this interval. ~ all of 30 observation: ( = (251 (3*50.06), (3*50.06)) = (100.82, ) So, given this interval, we include calories of donuts for all of the values. There are 30 donuts in this interval. Therefore, the empirical rule holds. Since, all three conditions are met. However, it does not mean that the data come from a mound shape (for example a normal distribution): We saw in the normal quantile plot, a value, 400 calories was away from the rest of the data. Therefore, the data is not normal. The mean is 251 and the median is 250. The data is about symmetric (or one can say slightly right skewed). The mode (most frequent number in the data) is 270. The mean, median, and mode are not all about the same or equal. Therefore, the data is not mound shape. And, furthermore, since a normal distribution is one type of mound shape, the data is not normal.