The Standard Deviation as a Ruler and the Normal Model Chapter 6 The Standard Deviation as a Ruler and the Normal Model

Topics The effect of the mean and standard deviation on the normal curve. Reading Table Z and find appropriate percentages or z-scores. The Empirical (68-95-99.7) Rule

Example Weekly payroll statistics for a small company. Lowest salary = $300 Mean Salary = $700 Median Salary = $500 Range = $1200 IQR = $600 First Quartile = $350 Standard Deviation = $400 Is the salary distribution symmetric or skewed? What is the third quartile? If everyone gets a $50 raise, how would this change all these stats? If everyone gets a 10% raise, how would this change these statistics?

Comparison Example Mary runs her Ski-to-Sea leg in 48 minutes. The mean is 52 minutes with a standard deviation of 6 minutes. Kim skis her leg in 38 minutes. The mean time is 42 minutes with a standard deviation of 7 minutes. Who performed better compared to the rest of the competition?

z-scores What does the z-score tell us?

Test Score example Open the file Test scores from the class web page. Make a histogram of the test scores? What is the distribution shape? Find the mean and standard deviation of the test scores. Standardize the test scores in the 3rd column. Make a histogram of the standardized scores. What is the distribution shape? Find the mean and standard deviation of the standardized scores.

Using the Standard Deviation as a Ruler Open the File Normal Curve Illustration from the course webpage Slide the mean and standard deviation. What is the effect of moving the mean? What is the effect of changing the standard deviation?

Empirical Rule (68-95-99.7 Rule) In a normal distribution, approximately 68% of the observations are within 1 standard deviation of the mean. 95% of the observations are within 2 standard deviations of the mean. 99.7% of the observations are within 3 standard deviations of the mean.

Determining the Mean and Standard Deviation from a Graph

Empirical Rule Application A sample of 200 high school students finds that the average number of hours per week spent studying (or doing homework) is 10.5. The standard deviation is 2.2 hours. The distribution is approximately normal. How many students (out of 200) study between 6.1 and 14.9 hours? At least 136 students (out of 200) study between _____ and ______ hours per week.

Example ex) The number of times a family in a small town eats out averages 2.3 times per month, with a standard deviation of 0.8 times per month. The distribution is approximately normal. Find the percentage of people that eat out less than 3 times per month. Find the percentage of people that eat out more than 1 time per month.

Using Table Z (Appendix E in the book) 2nd decimal place in z Z 0.00 0.01 0.02 0.03 0.04 0.0 0.5000 0.5040 0.5080 0.5120 0.5160 0.1 0.5398 0.5438 0.5478 0.5517 0.5557 0.2 0.5793 0.5832 0.5871 0.5910 0.5948

Using Fathom This is available on the course webpage under Fathom Data Files. It is called “Table T and Table Z Using Fathom.”

Using the TI-83/84 Under the DISTR key (2nd VARS), use [normalcdf]. The input pattern is normalcdf(lowerbound,upperbound[,μ,σ])

Example At a local road race, the average finishing time is 35 minutes with a standard deviation of 7 minutes. 80% of the finishers run faster than ____ minutes. 45% of the finishers run slower than____ minutes. Find the percentage of finishers running between 28 and 42 minutes.

Using Fathom

TI-83/84 For finding the z-score relating to a percentage (or area) to the left, use the InvNorm command. The sequence reads as InvNorm(area[,μ,σ])

Determining if Data is Normal We can use a Normal Quantile Plot in Fathom to determine if data is normal. If the plot appears approximately linear, the data is approximately normal.

Objectives Be able to find z-scores and interpret z-scores. Know how to use technology and Table z to find appropriate values. Understand and be able to utilize the empirical rule