Question

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 $y=high$temperature to $x=low$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 ${\mu }_y=16.6+1.02xwith\sigma =6.64°F$

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

Part a. $57.4°F$

Part b. $2.87\%$

Part c. No

Step-by-step solution

Part a. Step 1. Given information

${\mu }_y=16.6+1.02x$

$\sigma =6.64$

Part b. Step 2. Explanation

For the average high temperature $=40°F$

$$\begin{gathered} {\mu }_y=16.6+1.02x \\ =16.6+1.02\left(40\right) \\ =16.6+40.8 \\ =57.4 \end{gathered}$$

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

Part b. Step 1. Formula used

$z=\frac{x-\mu }{\sigma }$

Part b. Step 2. Explanation

Average mean

$$\begin{gathered} {\mu }_y=16.6+1.02x \\ =16.6+1.02\left(40\right) \\ =16.6+40.8 \\ =57.4 \\ \sigma =6.64 \end{gathered}$$

Z score is

$$\begin{gathered} z=\frac{x-\mu }{\sigma } \\ =\frac{70-57.4}{6.64} \\ =1.90 \end{gathered}$$

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

$$\begin{gathered} P(X>70)=P(Z>1.90) \\ =1-P(Z<1.90) \\ =1-0.9713 \\ =2.87\% \end{gathered}$$

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.