Question

PLEASE ANSWER IT!!! I NEED IT ASAP

Answer 1

a) y = 1882.48 + 91.03x

b) $91.03 billion per year

c) $3,338.96 billion

d) 2023

e) r = 0.993 (3 d.p.)

Very strong positive linear relationship, so as time increases, the total outstanding consumer credit tends to increase as well.

Explanation:

Part (a)

The least squares line (also known as the line of best fit or regression line), is a straight line that represents the best approximation of the linear relationship between two variables in a set of data points.

The equation of the least squares line is:

`\large\text{$y=a+bx$}`

where:

• y is the dependent variable (the variable we want to predict or explain).
• x is the independent variable (the variable used to predict y).
• a is the y-intercept of the line, which represents the value of y when x is zero.
• b is the slope of the line, which represents the rate of change of y with respect to changes in x.

As x is the number of years since 2000, the x and y values are:

`\begin{array}c\cline{1-6}x\;\sf(Year)&4&5&6&7&8\\\cline{1-6}y\; \sf(credit)&2250.5&2319.8&2437.0&2540.9&2595.1\\\cline{1-6}\end{array}`

We can use a statistical calculator to perform linear regression analysis on the given data. After entering the data into a statistical calculator we get:

`a = 1882.48`

`b = 91.03`

`r = 0.9934485954`

Therefore, the equation for the least squares line is:

`\large\boxed{y=1882.48+91.03x}`

where x is the number of years since 2000, and y is the outstanding consumer credit (in billions of dollars).

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Part (b)

The rate the consumer credit is growing per year is the slope of the least squares line equation found in part (a), so the value of b.

Therefore, the rate the consumer credit is growing is $91.03 billion per year.

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Part (c)

To use the result from part (a) to predict the amount of consumer credit in the year 2016, substitute x = 16 into the least squares line equation:

`\begin{aligned}y&=1882.48+91.03(16)\\y&=1882.48+1456.48\\y&=3338.96\end{aligned}`

Therefore, the predicted amount of consumer credit in the year 2016 is $3,338.96 billion.

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Part (d)

If this trend continues linearly, to find the year in which the total debt will first exceed $4000 billion, substitute y = 4000 into the equation from part (a) and solve for x:

`\begin{aligned}1882.48+91.03x&=4000\\91.03x&=2117.52\\x&=23.26178183...\end{aligned}`

As x is the number of years since 2000, the year in which the total debt will first exceed $4000 billion is 2023.

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Part (e)

The correlation coefficient (the r-value), is a statistical measure that quantifies the strength and direction of the linear relationship between two variables. It ranges between -1 and 1, where -1 indicates a perfect negative correlation, 0 indicates no correlation, and 1 indicates a perfect positive correlation.

In this case, the r-value is r = 0.993 (3 d.p.). This suggests that there is a very strong positive linear relationship between between the year and the total outstanding consumer credit. Therefore, as time increases, the total outstanding consumer credit tends to increase as well.