Question

4 omar has a piece of rope. he ties a knot in the rope and measures the new length of the rope. he then repeats this process several times. some of the data collected are listed in the table below.

number of knots	4	5	6	7	8
length of rope (cm)	64	58	49	39	31

state, to the nearest tenth, the linear regression equation that approximates the length, y, of the rope after tying x knots. explain what the y - intercept means in the context of the problem. explain what the slope means in the context of the problem.

Explanation:

Step1: Recall linear - regression formula

The linear - regression equation is of the form $y = mx + b$, where $m$ is the slope and $b$ is the y - intercept. We can use a calculator with linear - regression functionality (e.g., TI - 84 Plus: Stat, Edit to enter data, then Stat, Calc, LinReg(ax + b)). Let $x$ be the number of knots and $y$ be the length of the rope.

Step2: Calculate the slope and y - intercept

Using a calculator or statistical software, we find that the linear - regression equation is approximately $y=-10.5x + 106.1$.

Step3: Interpret the y - intercept

The y - intercept ($b = 106.1$) represents the length of the rope (in cm) when the number of knots $x = 0$. That is, the original length of the rope before tying any knots is approximately $106.1$ cm.

Step4: Interpret the slope

The slope ($m=-10.5$) represents the change in the length of the rope (in cm) for each additional knot tied. For every one additional knot tied, the length of the rope decreases by approximately $10.5$ cm.

Answer:

The linear - regression equation is $y=-10.5x + 106.1$. The y - intercept means the original length of the rope is approximately $106.1$ cm. The slope means that for each additional knot tied, the length of the rope decreases by approximately $10.5$ cm.