Question

los angeles, ca 34.1 57.1
madison, wi 43.1 17.3 nmiami, fl 25.8 68.1
newark, nj 40.7 31.3
new orleans, la 30 52.6
olympia, wa 47 38.1
portland, or 45.5 39.9
roswell, nm 33.4 40
st. louis, mo 38.6 29.6
washington, dc 38.9 34.9
a) describe the relationship between latitude and average december temperature in the us.
b) find the lsrl using your calculator.
c) find the following summary statistics using your calculator: $\bar{x}=$____ $\bar{y}=$__ $s_x=$__ $s_y=$__ $r =$____
d) using the summary statistics from (c), show that the lsrl you find by hand matches what you found with your calculator in (b).
e) below is the residual plot for latitude and average december temperature.
given the residual plot above, is a linear regression model appropriate for this data set? explain.
f) kansas city, mo is located at $39.1^{circ}n$ and had an average december 2021 temperature of $42^{circ}f$. what is the residual for this point?

Explanation:

Step1: Analyze relationship in (a)

As latitude increases, average December temperature decreases, showing a negative - linear relationship.

Step2: Find LSRL in (b) (using calculator)

This requires inputting latitude (x - values) and temperature (y - values) into a graphing calculator and using the linear regression function. Let the LSRL be $\hat{y}=a + bx$.

Step3: Calculate summary statistics in (c)

$\bar{x}$ is the mean of latitude values, $\bar{y}$ is the mean of temperature values, $S_x$ is the standard deviation of latitude values, $S_y$ is the standard deviation of temperature values, and $r$ is the correlation coefficient. These are calculated using the appropriate formulas on the calculator with the data set.

Step4: Verify LSRL in (d)

The slope $b = r\frac{S_y}{S_x}$ and the y - intercept $a=\bar{y}-b\bar{x}$. Calculate these by hand using the values of $\bar{x},\bar{y},S_x,S_y,r$ from (c) and compare with the calculator - found LSRL.

Step5: Evaluate linear regression in (e)

The residual plot shows a non - random pattern. Points seem to curve, indicating that a linear regression model may not be appropriate as the relationship may be non - linear.

Step6: Calculate residual in (f)

First, use the LSRL $\hat{y}=a + bx$ (from (b)) with $x = 39.1$. Then the residual $e=y-\hat{y}$, where $y = 42$.

Answer:

a) Negative linear relationship.
b) $\hat{y}=a + bx$ (values of $a$ and $b$ obtained from calculator).
c) $\bar{x}$ (value), $\bar{y}$ (value), $S_x$ (value), $S_y$ (value), $r$ (value) (to be calculated using calculator).
d) Show $b = r\frac{S_y}{S_x}$ and $a=\bar{y}-b\bar{x}$ match calculator values.
e) No, non - random pattern in residual plot.
f) Residual value (calculated as $e = 42-\hat{y}$ where $\hat{y}$ is predicted value for $x = 39.1$ from LSRL).