Lesson 4.1 Bivariate Data Today, we will learn to … > construct and use tables showing two different variables obtained from the same population

Bivariate Data The values of two different variables that are obtained from the same population. 1) both qualitative (attributes) 2) one qualitative - one quantitative 3) both quantitative (numerical)

TWO QUALITATIVE VARIABLES Examples? gender & favorite sport gender & favorite class cell phone & grade level

Senior Survey Activity Male Females Total Has a Car 95 78 173 No Car 81 68 149 176 146 322 What percentage of the teens have cars? How many students were surveyed? 173 322 322 54% What type of variables are used? What percentage of the boys had cars? 95 176 two qualitative variables 54%

ONE QUALITATIVE & ONE QUANTITATIVE Examples? Gender & Height Vehicle & Cost Vehicle & MPG

Guys & Height Girls & Height

TWO QUANTITATIVE VARIABLES Examples? age & height years employed & salary height & weight

The number of hours studied, x, compared to the grade earned, y 1 2 3 4 5 6 90 – 80 – 70 – 60 – number of hours spent studying Exam Grade The number of hours studied, x, compared to the grade earned, y

y x negative linear correlation

y x positive linear correlation

y x no linear correlation

y x no correlation

A marketing manager conducted a study to find out if there is a A marketing manager conducted a study to find out if there is a relation between money spent on advertising and sales. What do you predict?

Enter the data into List 1 and List 2 Advertising in 1000s Sales 2.4 225 1.6 184 2.0 220 2.6 240 1.4 180 186 2.2 215 Enter the data into List 1 and List 2

2nd Press Press Turn On Plot 1 Type: Scatter Plot ENTER Press Turn On Plot 1 Type: Scatter Plot Identify lists: L1 and L2 WINDOW Press and set dimensions Go to next slide

Advertising in 1000s Sales 2.4 225 1.6 184 2.0 220 2.6 240 1.4 180 186 2.2 215 There is a positive correlation between $$ spent on advertising and sales Xmin = _____ Xmax = _____ Xscl = _____ Ymin = _____ Ymax = _____ Yscl = _____ 3 1 150 250 25

correlation coefficient The correlation coefficient is a number r that represents the relationship between the two variables correlation coefficient –1 < r < 1

If r is close to –1, there is a strong negative correlation. This means as x increases, y decreases If r is close to 1, there is a strong positive correlation. This means as x increases, y also increases If r is close to 0, there is no linear correlation.

5 types of correlations Strong Positive Strong Negative Negative Positive Weak Negative Weak Positive No Correlation as “x” increases, “y” decreases as “x” increases, “y” increases What do they mean?

-1 -.9 -.8 -.7 -.6 -.5 -.4 -.3 -.2 -.1 0 .1 no correlation -1 -.9 -.8 -.7 -.6 -.5 -.4 -.3 -.2 Strong Negative Weak Negative Negative -.1 0 .1 no correlation .2 .3 .4 .5 .6 .7 .8 .9 1 Weak Positive Strong Positive Positive

Identify the type of correlation a) r = 0.81 Strong positive b) r = – 0.92 Strong negative c) r = 0.45 Weak positive d) r = 0.05 none e) r = – 0.35 Weak negative

Enter the two lists into two stat lists. Advertising in 1000s Sales 2.4 225 1.6 184 2.0 220 2.6 240 1.4 180 186 2.2 215 2nd Press Cursor to Diagnostic On ENTER Press twice

Press Cursor to CALC Choose 4: LinReg (ax+b) Identify lists L1 , L2 STAT Cursor to CALC Choose 4: LinReg (ax+b) Identify lists L1 , L2 ENTER Press

As x increases, y ___________ Correlation coefficient? r = 0.913 Type of correlation? Strong positive correlation increases As x increases, y ___________ Conclusion: As $$ spent on ads increases, sales ____________ increase

It's Time To Practice!

Lesson 4.2 Linear Regression Today, we will learn to… > write an equation that explains a linear correlation

n is the number of pairs of data We need to determine if our sample can be used to represent the entire population. Use the Critical Values Table n is the number of pairs of data

level of significance (α ) Using α = 0.05 means that we might be wrong ___% of the time 5 Using α = 0.01 means that we might be wrong __ % of the time. 1 Which is better? A 0.01 level of significance is better!!

Our study is significant. The sample represents the population n = 5 pairs of data r = - 0.893 level of significance level of significance α = 0.05 α = 0.01 critical value = 0.878 critical value = 0.959 | r | = _______ 0.893 0.893 > 0.878? 0.893 > 0.959? YES NO Our study is significant. The sample represents the population with a 5% error

n = 10 r = 0.950 | r | = 0.950 level of significance level of significance α = 0.05 α = 0.01 critical value = 0.632 critical value = 0.765 0.950 > 0.632? 0.950 > 0.765? YES YES Our study is significant. The sample represents the population with a 1% error

level of significance level of significance n = 7 r = - 0.750 | r | = 0.750 level of significance level of significance α = 0.05 α = 0.01 critical value = 0.754 critical value = 0.875 0.750 > 0.754? 0.750 > 0.875? NO NO Our study is not significant. The sample does not represents the population.

| r | > C.V. ? | r | > C.V. ? yes yes yes no no no α = 0.05 α = 0.01 | r | > C.V. ? | r | > C.V. ? yes yes The study is significant. The sample represents the population with a 1% error yes no The study is significant. The sample represents the population with a 5% error no no The study is NOT significant. The sample does not represent the population.

The fact that two variables are strongly correlated does not always prove a cause-and-effect relationship between the variables. 1) Does x cause y? 2) Should the variables be reversed? Does y cause x? 3) Could the relationship be caused by a third variable? 4) Could the relationship be a coincidence?

In many communities, there is a strong positive correlation between the amount of ice cream sold in a given month and the number of drownings that occur in that month. Does this mean that ice cream causes drowning? If not, what is an alternative explanation for the strong correlation?

It's Time To Practice!

Lesson 4.3 Linear Regression Today, we will learn to… > write an equation that explains a linear correlation

A linear regression line is a line of best fit for a scatter plot. The equation of a regression line is y = m x + b slope y-intercept

LinReg(ax+b) L1 , L2 a = – 4.3 b = 96.8 y = – 4x + 97 domain: Grade Absences Grade 1 95 7 65 3 80 2 85 5 77 4 70 93 75 82 LinReg(ax+b) L1 , L2 a = – 4.3 b = 96.8 y = – 4x + 97 domain: 1 < x < 7

You can use the equation to make predictions if the correlation between x and y is significant. For our example with absences and grades, the regression line is y = – 4x + 97 Predict the expected grade of someone with 1 absence. 93% - 4(1) + 97

It is not meaningful to predict the value of y for x = 25 because For our example with absences and grades, y = – 4x + 97 Predict the expected grade of someone with 25 absence. It is not meaningful to predict the value of y for x = 25 because 25 is outside the domain. Does this make sense? - 4(25) + 97 - 3%

The x-values must fall within the domain of the sample in order to use the equation to make predictions. IMPORTANT!

It's Time To Practice!