Chapter 4.4, Problem 3CYU

a.

To determine

The relationship between dependent and independent variable and make a scatter-plot for the given data set.

Answer to Problem 3CYU

The independent variable is the number of years and dependent variable is the median age of females when they are first married. There is positive correlation between the number of years and age.

Explanation of Solution

Given: The given data between the number of years and median age of females when they are first married is shown as

Number of years	Median age of females
1996	24.8
1997	25.0
1998	25.0
1999	25.1
2000	25.1
2001	25.1
2002	25.3
2003	25.3
2005	25.5
2006	25.9

Graph: The graph for the scatter-plot is shown as,

  

Conclusion: As the number of years increases, the age also increases. Therefore, there is a positive correlation between the number of years and median age of females when they are first married.

b.

To determine

The graph of line of fit for the scatter plot.

Explanation of Solution

Given: The given data between the number of years and median age of females when they are first married is shown as

Number of years	Median age of females
1996	24.8
1997	25.0
1998	25.0
1999	25.1
2000	25.1
2001	25.1
2002	25.3
2003	25.3
2005	25.5
2006	25.9

Graph: The line of fit for the scatter-plot is shown as, where number of years are represented on horizontal axis and median age of females is represented on vertical line.

  

c.

To determine

The equation for line of fit in slope intercept form.

Answer to Problem 3CYU

The equation for the line of fit is $y=0.11x-194.76$ .

Explanation of Solution

Given: The given data between the number of years and median age of females when they are first married is shown as

Number of years	Median age of females
1996	24.8
1997	25.0
1998	25.0
1999	25.1
2000	25.1
2001	25.1
2002	25.3
2003	25.3
2005	25.5
2006	25.9

Calculation: The slope $\left(m\right)$ for the given data can be calculated as,

  $\begin{array}{c}\text{Slope}\left(m\right)=\frac{25.9-24.8}{2006-1996}\\ =\frac{1.1}{10}\\ =0.11\end{array}$

Now, the line of fit having slope $\left(m=0.11\right)$ passes through the points $\left(1996,24.8\right)$ . Therefore, equation of line is written using point-slope form as,

  $\begin{array}{c}y-y_1=m\left(x-x_1\right)\mathrm{......}\left(\begin{array}{l}\text{Here }m=0.11\text{ and}\\ \left(x_1,y_1\right)\to \left(1996,24.8\right)\end{array}\right)\\ y-24.8=0.11\left(x-1996\right)\\ y=0.11x-\mathrm{194.76......}\left(1\right)\end{array}$

d.

To determine

The median age of females when they are first married in 2016.

Answer to Problem 3CYU

The median age of females will be 27 years.

Explanation of Solution

Given: The given data between the number of years and median age of females when they are first married is shown as

Number of years	Median age of females
1996	24.8
1997	25.0
1998	25.0
1999	25.1
2000	25.1
2001	25.1
2002	25.3
2003	25.3
2005	25.5
2006	25.9

Calculation: The median age of females in 2016 can be evaluated by plugging $\left(x=2016\right)$ in (1). Therefore, from (1),

  $\begin{array}{c}y=0.11\left(2016\right)-\mathrm{194.76......}\left(\text{For, }x=2016\right)\\ =\mathrm{27.......}\left(\text{Approx}\text{.}\right)\end{array}$

e.

To determine

Can the equation of fit give the reasonable estimate for the year 2056? Explain.

Answer to Problem 3CYU

Yes, the equation gives the reasonable estimate for the year 2056 because for the year 2056 the estimated age is 31.4 years.

Explanation of Solution

Given: The given data between the number of years and median age of females when they are first married is shown as

Number of years	Median age of females
1996	24.8
1997	25.0
1998	25.0
1999	25.1
2000	25.1
2001	25.1
2002	25.3
2003	25.3
2005	25.5
2006	25.9

Calculation: The median age of females in 2056 can be evaluated by plugging $\left(x=2056\right)$ in (1). Therefore, from (1),

  $\begin{array}{c}y=0.11\left(2056\right)-\mathrm{194.76......}\left(\text{For, }x=2056\right)\\ =\mathrm{31.4.......}\left(\text{Approx}\text{.}\right)\end{array}$

Hence, the equation of fit gives a reasonable estimate for the year 2056 because in the year 2056 the predicted age of women comes 31.4 years.