Answer to Question #268970 in Statistics and Probability for Light

Question

 the temperatures (in degrees Fahrenheit) at 18:00 hr. and the attendance (rounded to hundreds) at a minor league baseball team’s night games on 7 randomly selected evenings in May.

Temperature 61 70 50 65 48 78

Attendence 10 16 12 15 8 20

i. Do you think temperature depends on attendance or attendance depends on temperature? [2 marks]

ii. With temperature as an independent variable and attendance as a dependent variable, what is your hypothesis about the sign of b in the regression model?

[2 marks]

iii. Construct a scatter diagram for these data. Does the scatter diagram exhibit a

linear relationship between the two variables? [4 marks]

iv. Compute the correlation coefficient r and explain what it means. [5 marks]

v. Find the least squares regression line. Is the sign of b the same as the one you hypothesized for b in part ii? [5 marks]

vi. Use your regression equation to predict the attendance at a night game in May for a temperature of 60oF

Answer 1

i)

Attendance depends on temperature.

Temperature depends on climatic conditions.

ii)

b in the regression model has sign '+', because attendance (y) increases (from 8 to 20) with increasing (from 48 to 78) of temperature (x).

iii)

the scatter diagram does not exhibit a linear relationship between the two variables

iv)

correlation coefficient:

$r=\frac{\sum(x_i-\overline{x})(y_i-\overline{y})}{\sqrt{\sum(x_i-\overline{x})^2\sum(y_i-\overline{y})^2}}=0.8974$

This indicates that there is a significant large positive relationship between x and y.

v)

equation of regression line:

$y=bx+a$

where

$b=\frac{n\sum xy-\sum x\sum y}{n\sum x^2-(\sum x)^2}=0.3388$

$a=\frac{\sum y\sum x^2-\sum x\sum xy}{n\sum x^2-(\sum x)^2}=-7.506$

$y=0.3388x-7.506$

the sign of b is the same as in the hypothesis in part ii)

vi)

the attendance for a temperature of 60oF:

$y=0.3388\cdot60-7.506=12.822\approx 13$