Question

A study is given in which scientists examined data on mean sea surface temperatures (in degrees Celsius) and mean coral growth (in millimeters per year) over a several-year period at different locations. Here are the data:

Sea Surface Temperature    29.69    29.86    30.16    30.21    30.48    30.65    30.90

Growth                                  2.62      2.57      2.68      2.60      2.49      2.39      2.25

(a) Use your calculator to find the mean and standard deviation of both sea surface temperature x and growth y and the correlation r between x and y. Use these basic measures to find the equation of the least-squares line for predicting y from x. (Round your answers to three decimal places.)
ŷ = _____ + _____x

(b) Enter the data into your software or calculator, and use the regression function to find the least-squares line. The result should agree with your work in part (a) up to roundoff error. (Round your answers to three decimal places.)
ŷ =______ +_____x

(c) Say in words what the numerical value of the slope tells you. (Round your answer to three decimal places.)

Every increase of one degree Celsius means about ______ fewer mean millimeters of coral growth per year.

Answer 1

We will use R studio to answer the above question . Codes are marked in Bold

(a)

# x= Sea Surface Temperature, y = Growth

x<- c(29.69,29.86,30.16,30.21,30.48,30.65,30.90)
y <- c(2.62,2.57,2.68,2.60,2.49,2.39,2.25)

# x_bar =mean of x , y_bar = mean of y

x_bar <- mean(x)
y_bar <- mean(y)

# sx = Standard deviation of x, sy = standard deviation of y

sx <- sd(x)
sy <- sd(y)

# r = correlation between x &y

r <- cor(x,y)

# slope

slope <- (r*sy)/(sx)

# intercept

intercept <- y_bar-slope*x_bar

print(round(x_bar,3))

print(round(y_bar,3))
print(round(sx,3))

print(round(sy,3))

print(round(r,3))

print(round(slope,3))

print(round(intercept,3))

Mean of x = 30.279

Mean of y = 2.514

standard deviation of x = 0.429

standard deviation of y =0.15

correlation between x & y =  -0.845

Slope of y on x = -0.295

intercept = 11.456

The regression equation

y = -0.295*x + 11.456

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(b)

# (b)
reg_equation <- lm(y~x)
summary(reg_equation)

Residuals:
1 2 3 4 5 6 7
-0.06810 -0.06790 0.13070 0.06546 0.03520 -0.01459 -0.08076

Coefficients:
Estimate Std. Error t value Pr(>|t|)   
(Intercept) 11.45624 2.53064 4.527 0.00624 **
x -0.29532 0.08357 -3.534 0.01667 *
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.08785 on 5 degrees of freedom
Multiple R-squared: 0.7141,   Adjusted R-squared: 0.6569
F-statistic: 12.49 on 1 and 5 DF, p-value: 0.01667

Hence, the equation y = -0.295*x + 11.456

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(c)

Every increase of one degree Celsius means about -0.295 fewer mean millimeters of coral growth per year.