Chapter 12: Q. 2 (page 787)

Using the daily high and low temperature readings at Chicago’s O’Hare International Airport for an entire year, a meteorologist made a scatterplot relating $y = h i g h$ temperature to $x = l o w$ temperature, both in degrees Fahrenheit. After verifying that the conditions for the regression model were met, the meteorologist calculated the equation of the population regression line to be $μ_{y} = 16.6 + 1.02 x w i t h σ = 6.64 ° F$
a. According to the population regression line, what is the average high temperature on days when the low temperature is $40 ° F$ ?
b. About what percent of days with a low temperature of 40°F have a high temperature greater than $70 ° F$ ?
c. If the meteorologist used a random sample of $10$ days to calculate the regression line instead of using all the days in the year, would the slope of the sample regression line be exactly $1.02$ ? Explain your answer.

Short Answer

Expert verified

Part a. $57.4 ° F$

Part b. $2.87 %$

Part c. No

Step by step solution

Part a. Step 1. Given information

$μ_{y} = 16.6 + 1.02 x$

$σ = 6.64$

Part b. Step 2. Explanation

For the average high temperature $= 40 ° F$

$μ_{y} = 16.6 + 1.02 x = 16.6 + 1.02 (40) = 16.6 + 40.8 = 57.4$

Therefore the average high temperature as per the population regression line is $57.4 ° F$

Part b. Step 1. Formula used

$z = \frac{x - μ}{σ}$

Part b. Step 2. Explanation

Average mean

$μ_{y} = 16.6 + 1.02 x = 16.6 + 1.02 (40) = 16.6 + 40.8 = 57.4 σ = 6.64$

Z score is

$z = \frac{x - μ}{σ} = \frac{70 - 57.4}{6.64} = 1.90$

Corresponding probability for the greater than $70 ° F$ .

$P (X > 70) = P (Z > 1.90) = 1 - P (Z < 1.90) = 1 - 0.9713 = 2.87 %$

Therefore about $2.87$ percent of the days with a low temperature of $40 ° F$ are expected to being having high temperature that is greater than $70 ° F$ .

Part c. Step 1. Explanation

No, the reason is that the slope of the population regression line is $1.02$ and it is predicted that the slope of the regression line of sample is very near to $1.02$ but it is not exactly so there would be some sampling variability in a sample.