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Marriage Prospects Data released by the Census Bureau in 1986 indicated the likelihood that never-married women would eventually marry. The data indicated that the older the woman, the less the likelihood of marriage. Specifically, two statistics indicated that women who were 45 and never-married had an 18 percent chance of marriage and women 25 years old had a 78 percent chance of marriage. Assume that a linear fit to these two data points provides a reasonable approximation for the function p=f(a), where p equals the probability of marriage and a equals the age of a never- married woman.
(a) Determine the linear function p=f(a).
(b) Interpret the slope and p intercept.
(c) Do the values in part b seem reasonable?
(d) If the restricted domain on this function is 20 sa s 50, determine f(20), f(30), f(40), and f(50).

Respuesta :

MrRoyal

Answer:

a. [tex]f(a) = -0.03a +1.53[/tex]

b. See Explanation

c. The slope is reasonable but the p intercept is not

d. [tex]f(20) = 93\%[/tex]     [tex]f(30) = 63\%[/tex]    [tex]f(40) = 33\%[/tex]   [tex]f(50) = 3\%[/tex]

Step-by-step explanation:

Given

[tex]a = age[/tex]

[tex]p = probability\ of\ marriage[/tex]

[tex]a = 45[/tex] when [tex]p = 18\%[/tex]

[tex]a = 25[/tex] when [tex]p = 78\%[/tex]

Solving (a): The linear function

We start by calculating the slope, m

[tex]m = \frac{p_2 - p_1}{a_2 - a_1}[/tex]

[tex]m = \frac{78\% - 18\%}{25- 45}[/tex]

[tex]m = \frac{60\%}{-20}[/tex]

[tex]m = -3\%[/tex]

[tex]m = -0.03[/tex]

The function is then calculated as follows

[tex]p - p_1 = m(a - a_1)[/tex]

This gives:

[tex]p - 18\% = -0.03(a - 45)[/tex]

[tex]p - 0.18 = -0.03(a - 45)[/tex]

[tex]p - 0.18 = -0.03a +1.35[/tex]

Solve for p

[tex]p= -0.03a +1.35+0.18[/tex]

[tex]p= -0.03a +1.53[/tex]

Hence,

[tex]f(a) = -0.03a +1.53[/tex]

Solving (b): Interpret the slope and the p intercept

The slope is calculated as:

[tex]m = -0.03[/tex]

And it implies that, there is a 3% reduction in change of getting older as women get older

The p intercept implies that, there is a 1.53 chance for 0 years old female child to get married.

Solving (c): Is (b) reasonable

The slope is reasonable.

However, the p intercept is not because of the age of the woman

Solving (d): Determine f(20), f(30), f(40), f(50)

We have that:

[tex]f(a) = -0.03a +1.53[/tex]

[tex]f(20) = -0.03 * 20 + 1.53[/tex]

[tex]f(20) = -0.6 + 1.53[/tex]

[tex]f(20) = 0.93[/tex]

[tex]f(20) = 93\%[/tex]

[tex]f(30) = -0.03 * 30 + 1.53[/tex]

[tex]f(30) = -0.9 + 1.53[/tex]

[tex]f(30) = 0.63[/tex]

[tex]f(30) = 63\%[/tex]

[tex]f(40) = -0.03 * 40 + 1.53[/tex]

[tex]f(40) = -1.2 + 1.53[/tex]

[tex]f(40) = 0.33[/tex]

[tex]f(40) = 33\%[/tex]

[tex]f(50) = -0.03 * 50 + 1.53[/tex]

[tex]f(50) = -1.5 + 1.53[/tex]

[tex]f(50) = 0.03[/tex]

[tex]f(50) = 3\%[/tex]

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