Question

1. the table shows the number of active woodpecker clusters in a part of the de soto national forest in mississippi.

year|1992|1993|1994|1995|1996|1997|1998|1999|2000
active clusters|22|24|27|27|34|40|42|45|51
a) find a linear equation that models the data.
b) what does the domain represent?
c) what does the range represent?
d) interpret the rate of change.
e) use the equation to determine the number of active clusters in the year 2010

1. the table shows the duration of several eruptions of the geyser old faithful and interval between eruptions.

duration (minutes)|1.5|2.0|2.5|3.0|3.5|4.0|4.5|5.0
interval(minutes)|50|57|65|71|76|82|89|95
a) find the regression equation.
b) interpret the rate of change.
c) predict the interval between geysers for a duration of 6 minutes.

Explanation:

Step1: Define variables for woodpecker data

Let $x$ be the number of years since 1992. So for 1992, $x = 0$; for 1993, $x=1$ and so on. Let $y$ be the number of active wood - pecker clusters.
We have the points $(0,22)$ and $(8,51)$ (since for 2000, $x = 8$).

Step2: Calculate the slope $m$

The slope formula is $m=\frac{y_2 - y_1}{x_2 - x_1}$. Using the points $(0,22)$ and $(8,51)$:
$m=\frac{51 - 22}{8-0}=\frac{29}{8}=3.625$

Step3: Find the y - intercept $b$

The equation of a line is $y=mx + b$. Using the point $(0,22)$ (where $x = 0$ and $y = 22$), we substitute into the equation:
$y=mx + b\Rightarrow22=3.625\times0 + b$, so $b = 22$.
The linear equation is $y = 3.625x+22$.

Step4: Answer part b

The domain represents the set of years. In terms of our $x$ - variable (years since 1992), the domain is the set of non - negative integers $x = 0,1,\cdots,8$ and can be extended for future predictions. In the context of the real - world situation, it represents the years from 1992 to 2000 and potentially beyond for prediction purposes.

Step5: Answer part c

The range represents the number of active woodpecker clusters. It is the set of values $\{22,24,27,34,40,42,45,51\}$ and values predicted by the linear model for other years in the domain or for future years.

Step6: Interpret the rate of change

The rate of change (slope) $m = 3.625$ means that, on average, the number of active woodpecker clusters increases by 3.625 each year.

Step7: Answer part e

For the year 2010, $x=2010 - 1992=18$.
Substitute $x = 18$ into the equation $y=3.625x + 22$:
$y=3.625\times18+22=65.25+22=87.25\approx87$

For the geyser data:

Step1: Calculate the regression equation

We use the least - squares regression formula for a line $y=mx + b$.
First, we calculate the following sums:
Let $x$ be the duration and $y$ be the interval.
$n = 8$ (number of data points)
$\sum x=1.5 + 2.0+2.5+3.0+3.5+4.0+4.5+5.0=26$
$\sum y=50 + 57+65+71+76+82+89+95=585$
$\sum x^2=1.5^2+2.0^2+2.5^2+3.0^2+3.5^2+4.0^2+4.5^2+5.0^2=96.5$
$\sum xy=1.5\times50+2.0\times57+2.5\times65+3.0\times71+3.5\times76+4.0\times82+4.5\times89+5.0\times95=2035$

The slope $m=\frac{n\sum xy-\sum x\sum y}{n\sum x^2-(\sum x)^2}$
$m=\frac{8\times2035 - 26\times585}{8\times96.5-26^2}=\frac{16280-15210}{772 - 676}=\frac{1070}{96}\approx11.146$

The y - intercept $b=\frac{\sum y-m\sum x}{n}$
$b=\frac{585-11.146\times26}{8}=\frac{585 - 289.8}{8}=\frac{295.2}{8}=36.9$

The regression equation is $y = 11.146x+36.9$

Step2: Interpret the rate of change

The slope $m = 11.146$ means that for every one - minute increase in the duration of an eruption of the geyser Old Faithful, the interval between eruptions increases by approximately 11.146 minutes.

Step3: Predict the interval for a 6 - minute duration

Substitute $x = 6$ into the equation $y=11.146x+36.9$
$y=11.146\times6+36.9=66.876+36.9=103.776\approx104$

Answer:

a) For woodpecker data: $y = 3.625x+22$; For geyser data: $y = 11.146x+36.9$
b) For woodpecker data: The set of years (years since 1992 in our model), representing the time period under study and potentially for prediction.
c) For woodpecker data: The number of active woodpecker clusters.
d) For woodpecker data: On average, the number of active woodpecker clusters increases by 3.625 each year; For geyser data: For every one - minute increase in eruption duration, the interval between eruptions increases by approximately 11.146 minutes.
e) For woodpecker data: Approximately 87; For geyser data: Approximately 104