Section 2.2 Standard Normal Calculations AP Statistics September 17th , 2014

AP Statistics, Section 2.1, Part 2 Comparing data sets How do we compare results when they are measured on two completely different scales? One solution might be to look at percentiles What might you say about a woman that is in the 50th percentile and a man in the 15th percentile? AP Statistics, Section 2.1, Part 2

Another way of comparing Another way of comparing: Look at whether the data point is above or below the mean, and by how much. Example: A man is 64 inches tall. The heights of men are normally distributed with a mean of 69 inches and standard deviation of 2.5 inches. AP Statistics, Section 2.1, Part 2

Another way of comparing Example: A man is 64 inches tall. The heights of men are normally distributed with a mean of 69 inches and standard deviation of 2.5 inches. We can see that the man is below the mean, but by how much? AP Statistics, Section 2.1, Part 2

Another way of comparing Example: A man is 64 inches tall. The heights of men are normally distributed with a mean of 69 inches and standard deviation of 2.5 inches. AP Statistics, Section 2.1, Part 2

AP Statistics, Section 2.1, Part 2 z-scores The z-score is a way of looking at every data set, because each data set has a mean and standard deviation We call the z-score the “standardized” score. AP Statistics, Section 2.1, Part 2

AP Statistics, Section 2.1, Part 2 z-scores Positive z-scores mean the data point is above the mean. Negative z-scores mean the data point is below the mean. The larger the absolute value of the z-score, the more unusual it is. AP Statistics, Section 2.1, Part 2

AP Statistics, Section 2.1, Part 2 Using the z-table We can use the z-table to find out the percentile of the observation. A z-score of -2.0 is at the 2.275 percentile. AP Statistics, Section 2.1, Part 2

AP Statistics, Section 2.1, Part 2 Cautions The z-table only gives the amount of data found below the z-score. If you want to find the portion found above the z-score, subtract the probability found on the table from 1. AP Statistics, Section 2.1, Part 2

Standardized Normal Distribution We should only use the z-table when the distributions are normal, and data has been standardized N(μ,σ) is a normal distribution N(0,1) is the standard normal distribution “Standardizing” is the process of doing a linear translation from N(μ,σ) into N(0,1) AP Statistics, Section 2.1, Part 2

AP Statistics, Section 2.1, Part 2 Example Men’s heights are N(69,2.5). What percent of men are taller than 68 inches? AP Statistics, Section 2.1, Part 2

AP Statistics, Section 2.1, Part 2 Step 1 State the problem in terms of x, an observed variable. x is the height 68 inches. AP Statistics, Section 2.1, Part 2

AP Statistics, Section 2.1, Part 2 Step 2 Standardize x to restate the problem in terms of a standard normal distribution. AP Statistics, Section 2.1, Part 2

AP Statistics, Section 2.1, Part 2 Step 3 Find the required area under the standard normal curve. Remember, the total area is 1. AP Statistics, Section 2.1, Part 2

AP Statistics, Section 2.1, Part 2 Step 4 Write your conclusion in the context of the problem. About 65.542% of the population of men are taller than 68 inches. AP Statistics, Section 2.1, Part 2

Working with intervals What proportion of men are between 68 and 70 inches tall? AP Statistics, Section 2.1, Part 2

Working with intervals AP Statistics, Section 2.1, Part 2

Working with intervals What proportion of men are between 68 and 70 inches tall? AP Statistics, Section 2.1, Part 2

AP Statistics, Section 2.1, Part 2 Assignment Exercises 2.19 – 2.25, The Practice of Statistics. AP Statistics, Section 2.1, Part 2