Question

At what age do babies learn to crawl? Does it take longer to learn in the winter, when babies are often bundled in clothes that restrict their movement? Perhaps there might even be an association between babies’ crawling age and the average temperature during the month they first try to crawl (around $6$months after birth). Data were collected from parents who brought their babies to the University of Denver Infant Study Center to participate in one of a number of studies. Parents reported the birth month and the age at which their child was first able to creep or crawl a distance of $4$feet within one minute. Information was obtained on localid="1663582611977" $414$infants (localid="1663582615119" $208$boys and localid="1663582618211" $206$girls). Crawling age is given in weeks, and average temperature (in degrees Fahrenheit) is given for the month that is localid="1663582621791" $6$months after the birth month.

a. Make an appropriate graph to display the relationship between average temperature and average crawling age. Describe what you see. Some computer output from a least-squares regression analysis of the data is shown.

b. What is the equation of the least-squares regression line that describes the relationship between average temperature and average crawling age? Define any variables that you use.

c. Interpret the slope of the regression line.

d. Can we conclude that warmer temperatures $6$months after babies are born causes them to crawl sooner? Justify your answer.

Short answer

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

Part b. Equation for least - square regression line, $\hat{y}=35.68-0.0777x$

Part c. On average, the average crawling age drops by an average temperature of $0.0777$weeks per $F°$

Part d. No, the statement cannot be concluded.

Step-by-step solution

Part (a) Step 1. Given information

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

Part (a) Step 2. Explanation

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.

Part (b) Step 1. Explanation

Least – squares regression line (general equation):

$\hat{y}=b_0+b_{1}x$

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:

role="math" localid="1663583182993" $\hat{y}=b_0+b_{1}x=35.68-0.0777x$

Where,

x: average temperature

y: average crawling age

Part (c) Step 1. Explanation

From Part (b),

Least – squares regression line general equation:

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

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$°F$.

Part (d) Step 1. Explanation

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.