Question

Question 1

The average sea surface temperature (in degree Celsius) and the coral growth (in millimetres per year) over 18 years at a particular location are observed. An excerpt of the data is shown in the table below:

Temperature (x)= 29.61 29.82 30.25 … 30.96

Growth (y)= 2.63 2.58 2.49 … 2.26

The output from R Commander appears on the next page

(c) Using the R Commander output, calculate the correlation coefficient between the two variables. [2 marks]

(d) Test the significance of the slope of the linear regression line at the 5% level of significance. State clearly the null and alternative hypotheses, the name of the test or the test statistic, decision rule, test result and conclusion in terms of the original problem. [6 marks]

(e) Write down the linear regression equation for the data. (1MARK)

(f) Interpret the slope of the linear regression equation. [1 mark]

(g) Use the equation in (e), predict the coral growth when the average sea surface temperature is 30 degree Celsius. [1 mark]

(h) Comment on the appropriateness of the prediction in (g). [2 marks]

(i) Write down three assumptions underlying the analysis. [3 marks]

> summary(RegModel.1)

Call: lm(formula = Growth ~ Temperature, data = Q1)

Residuals:

Min 1Q Median 3Q Max

-0.10279 -0.04107 -0.01688 0.04901 0.09864

Coefficients:

Estimate Std. Error t value Pr(>|t|)

(Intercept) 7.81986 1.11135 7.036 2.81e-06 ***

Temperature -0.17860 0.03698 -4.830 0.000185 ***

---

Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 0.06155 on 16 degrees of freedom

Multiple R-squared: 0.5931, Adjusted R-squared: 0.5677

F-statistic: 23.33 on 1 and 16 DF, p-value: 0.0001848

Answer 1

c) By using The R commander output.

R square is =0.5931

Hence the correlation coefficient is =sqrt(0.5931)

r=-0.77

## Since the estimate of slope indicates the negative linear relationship between temprature and growth.

d) The hypothesis testing problem is

Vs

The test statistics is:

t=-4.830

Corresponding pvalue is P=0.000185

Since pvalue is less than 0.05 we reject the null hypothesis.

Therefore the slope of the linear regression line is significant in the regression model.

Therefore The average sea surface temperature is useful in predicting the coral growth.

e) The linear regression model is:

f) The slope of regression model indicates that the increase in average sea surface temperature by one degree celsius result in the decrease of coral growth by 0.178 millimetres per year.

g) The coral growth when the average sea surface temperature is 30 degree Celsius is given by:

The coral growth when the average sea surface temperature is 30 degree Celsius is 2.46186

g) Since the value of average sea surface temperature that is 30 degree Celsius is interpolated value,

the regression line best predict the value of the coral growth.