Bell Ringer Make a scatterplot for the following data. (in the calculator)

Chapter 10 Notes The association between year (measured in years since 1800) and U.S. population (in millions) is strong, positive, and curved. We cannot use a regression line to model this relationship without re-expressing the data first.

If the association of two variables is curved, we can’t use linear regression for the data, but we can attempt to re-express the data by one of a few methods: Square it: 𝑦 2 Square root it: 𝑦 Take the log of one or both variables: log 𝑦 𝑜𝑟 log𝑥 Take the reciprocal: 1 𝑦

The scatterplot of sqrt(population) and year is still a bit curved, but straight enough to fit a line. The model is 𝑃𝑜𝑝 =1.46275+.07457(𝑌𝑒𝑎𝑟)

How to do this on your calculator: Enter the original data in L1 and L2 Instead of going through every single entry and recalculating it, use L3 For this example, make L3 = 𝐿2 Do linear regression using L1 & L3

log 𝑦 =𝑥 → 10 𝑥 =𝑦 Reminder of how logarithms work Examples: 10 2.73 = 𝑦 𝑦 =537.03 𝑦 =log⁡(345) In calculator, type log(345) 𝑦 =2.54

Reminder of how logarithms work ln 𝑦 =𝑥 → 𝑒 𝑥 =𝑦 Example: ln 𝑦 =7.8 𝑒 7.8 = 𝑦 𝑦 =2440.6

The scatterplot of years since 1970 and mortgage rates in millions is curved. The residual plot is shown on the right. Year Millions 1970 1.2 1972 2.5 1974 2.9 1976 3.1 1978 5.8 1980 8.3 1982 10.8 1984 14.7 1986 21.8 1988 29.7

A linear model is not appropriate for the scatterplot of years since 1970 and mortgage rates, but it can be straightened using log(y). Year Millions 1970 1.2 1972 2.5 1974 2.9 1976 3.1 1978 5.8 1980 8.3 1982 10.8 1984 14.7 1986 21.8 1988 29.7 You Try: Create a model using years since 1970 and log(millions). Use the model to estimate the mortgage rate in 1981.

Scatterplot for years since 1970 and log(mortgage) Residual plot for years since 1970 and log(mortgage)

The appropriate model is: log 𝑚𝑜𝑟𝑡𝑔𝑎𝑔𝑒 =0.1462+.074(𝑦𝑒𝑎𝑟𝑠 𝑠𝑖𝑛𝑐𝑒 1970) The estimate for the year 1981 using this model is: log 𝑚𝑜𝑟𝑡𝑔𝑎𝑔𝑒 =0.1462+.074 11 log 𝑚𝑜𝑟𝑡𝑔𝑎𝑔𝑒 =0.9602 𝑚𝑜𝑟𝑡𝑔𝑎𝑔𝑒 =10 .9602 =9.12 𝑚𝑖𝑙𝑙𝑖𝑜𝑛

Practice: Use re-expressed data to create a model that predicts ticket prices (scale the year). What is the predicted ticket price for 2004? Answer: $11.67

Homework for Chapters 9 & 10 Read chapters 9 & 10 From Monday: p.215 #1-3, 9, 12 Add to HW: p.215 #20; p.239 #5, 6, 15, 16