Chapter 5.2, Problem 59E

(a)

To determine

Appropriate graph to display the relationship between average temperature and average crawling age and discuss the observations.

Answer to Problem 59E

Linear, negative, and moderately strong association with one outlier.

Explanation of Solution

Given information:

Information obtained for the month that is 6 months after the birth month:

  

Scatterplot for average temperature and average crawling age relationship:

On horizontal axis,

Average temperature has been represented.

On vertical axis,

Average crawling age has been represented.

  

Form: Linear, because in the scatterplot no strong curvature exists.

Strength: Moderately strong, because the points in the scatterplot are not spread far apart and do not lie extremely close together.

Unusual features: It has been shown that the leftmost in the scatterplot seems to deviate from the general linear pattern in the other points; one outlier appears to be there.

Direction: Negative, the pattern slops downward in the scatterplot.

This implies

There is a negative, linear, moderately strong association between the average temperature and the average crawling age, an outlier also exists in the scatterplot.

(b)

To determine

Equation of the least − square regression line describing the relationship between average temperature and average crawling age.

Answer to Problem 59E

Equation for least - square regression line,

  $\widehat{y}=35.68-0.0777x$

Explanation of Solution

Given information:

Information obtained for the month that is 6 months after the birth month:

  

Least − squares regression line (general equation):

  $\widehat{y}=b_0+b_{1}x$

From computer output,

Slope $b_1$ estimate has been provided in row “Average temperature” and column “Coef”:

  $b_1=-0.0777$

And

Constant $b_0$ estimate has been provided in row “Constant” and column “Coef”:

  $b_0=35.68$

Substitute the values of $b_1$ and $b_0$ in the general equation of the least − square regression line:

  $\widehat{y}=b_0+b_{1}x=35.68-0.0777x$

Where,

x : average temperature

y : average crawling age

(c)

To determine

Interpretation of slope of regression line.

Answer to Problem 59E

On average, the average crawling age drops by an average temperature of 0.0777 weeks per $\text{F}$ .

Explanation of Solution

Given information:

Information obtained for the month that is 6 months after the birth month:

  

From Part (b),

Least − squares regression line general equation:

  $\widehat{y}=b_0+b_{1}x=35.68-0.0777x$

Where,

x : average temperature

y : average crawling age

Now,

In the least square regression equation,

The slope is the coefficient of x .

And

Also represents the average increase (or decrease) of y per unit of x .

Thus,

On average, the average crawling age drops by an average temperature of 0.0777 weeks per $\text{F}$ .

(d)

To determine

Whether it can be concluded that warmer temperature 6 months after babies are born causes them to crawl sooner.

Answer to Problem 59E

No, the statement cannot be concluded.

Explanation of Solution

Given information:

Information obtained for the month that is 6 months after the birth month:

  

From Part (a),

We came to know that

There was a linear, negative, moderately strong association between the average temperature and average crawling age.

However,

This association does not mean causation.

Which means

The increasing average temperature does not necessarily cause a decrease in average crawling age.

Thus,

We cannot conclude that warmer temperature 6 months after babies are born causes them to crawl sooner.