Question

A nuclear power plant releases water into a nearby lake every afternoon at $4:51$p.m. Environmental researchers are concerned that fish are being driven away from the area around the plant. They believe that the temperature of the water discharged may be a factor. The scatterplot below shows the temperature of the water (°C) released by the plant and the measured distance (in meters) from the outflow pipe of the plant to the nearest fish found in the water on eight randomly chosen afternoons.

Computer output from a least-squares regression analysis on these data is shown below.

(a) Write the equation of the least-squares regression line. Define any variables you use.

(b) Interpret the slope of the regression line in context.

(c) A residual plot for the regression is shown on the next page. Is a linear model appropriate for describing the relationship between temperature and distance to the nearest fish? Justify your answer.

(d) Does the linear model in part (a) overpredict or underpredict the measured distance from the outflow pipe to the nearest fish found in the water for a temperature of $29°$? Explain your reasoning.

Short answer

(a) The equation least square regression line is $y=-73.64+5.7188x$

(b) The slope of the equation is $5.7188$.

(c)The linear model is appropriate for the graph.

(d)The residual value is negative so predicted value will be high.

Step-by-step solution

Part (a) Step 1: Given Information

We are given that A nuclear power plant releases water into a nearby lake every afternoon at $4:51$p.m. and we have to find out the equation of least square regression line.

Part (a) Step 2: Explanation

According to the question,

constant of the equation is the intercept and temperature is the slope,

which results,

$y=-73.64+5.7188x$where x is temperature

Hence, this is the required equation.

Part (b) Step 1: Given Information

We are given with the equation that we have find out in the part (a) and we have to interpret the slope.

Part (b) Step 2: Explanation

From the equation,

we get $slope=5.7188$

Now according to the problem for every $1°$Celsius, the fish are $5.7188m$away from the outflow pipe of the plant.

Part (c) Step 1: Given Information

We are given with the residual plot for the regression and we have to find out if a linear model appropriate for describing the relationship between temperature and distance to the nearest fish.

Part (c) Step 2: Explanation

Now, we will look for any pattern that can violate the residual assumptions but there does not appear any pattern So, the residual assumption is not not violated.

Hence, a linear model is appropriate for describing the relationship between temperature and distance to the nearest fish.

Part (d) Step 1: Given Information

We have to find out if the linear model in part (a) overpredict or underpredict the measured distance from the outflow pipe to the nearest fish found in the water for a temperature of $29°$.

Part (d) Step 2: Explanation

Now, from the equation in Part (a)

which is $y=-73.64+5.7188x$

we will find the value for temperature $29°$

which gives $y=-73.64+5.7188*29=93.2052$

Now check the value of y in the residual plot which gives that

residual is about $-15to-18$.

Hence, the residual is negative the value will also be high.