You have collected samples of chlorophyll content from various locations off the NSW coast and obtained satellite imagery from those same locations and sample times. The in situ measured chlorophyll content is labelled as HPLC and your satellite information is labelled as OC3. You have run a simple linear regression (assumptions seemed good) and have obtained the output below. From the output, can we conclude Ocean colour is a significant predictor of surface chlorophyll content?

Call:
lm(formula = HPLC ~ OC3, data = OC)

Residuals:
     Min       1Q   Median       3Q      Max
-0.42257 -0.05749 -0.00551 0.07614 0.30317

Coefficients:
            Estimate Std. Error t value Pr(|t|)
(Intercept) -0.22195    0.07816 -2.840 0.00981 **
OC3          1.90465    0.26480   7.193 4.34e-07 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.158 on 21 degrees of freedom
Multiple R-squared: 0.7113, Adjusted R-squared: 0.6975
F-statistic: 51.74 on 1 and 21 DF, p-value: 4.337e-07

Question

You have collected samples of chlorophyll content from various locations off the NSW coast and obtained satellite imagery from those same locations and sample times.  The in situ measured chlorophyll content is labelled as HPLC and your satellite information is labelled as OC3. You have run a simple linear regression (assumptions seemed good) and have obtained the output below. From the output, can we conclude Ocean colour is a significant predictor of surface chlorophyll content?

Call:
lm(formula = HPLC ~ OC3, data = OC)

Residuals:
     Min       1Q   Median       3Q      Max
-0.42257 -0.05749 -0.00551  0.07614  0.30317

Coefficients:
            Estimate Std. Error t value Pr(>|t|)   
(Intercept) -0.22195    0.07816  -2.840  0.00981 **
OC3          1.90465    0.26480   7.193 4.34e-07 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.158 on 21 degrees of freedom
Multiple R-squared:  0.7113,  Adjusted R-squared:  0.6975
F-statistic: 51.74 on 1 and 21 DF,  p-value: 4.337e-07

BlackTom AI · Accepted Answer

We start by restating the key information available from the regression output to ground the analysis.
The model is: HPLC ~ OC3, with intercept and OC3 coefficient estimates provided, along with their standard errors, t-values, and p-values. The overall model statistics include a Residual standard error of 0.158 on 21 degrees of freedom, a Multiple R-squared of 0.7113, an Adjusted R-squared of 0.6975, and an F-statistic of 51.74 with p-value 4.337e-07.
Now, evaluate each answer option in light of these quantities:
Option A: "We cannot make such a conclusion from the above summary, more information is needed." This is overly cautious. The given output includes a p-value for OC3 of 4.34e-07, which is far below any conventional alpha level (e.g., 0.05). This p-value directly tests the null hypothesis that the slope for OC3 is zero, i.e., that OC3 does not predict HPLC. Given such a small p-value, there is strong evidence against the null, so concluding that OC3 is a significant predictor is justified. The high t-value (7.193) and the substantial F-statistic further corroborate this.
Option B: "As P < 0.05, we can conclude OC3 (Ocean Colour) is a significant predictor of HPLC (chlorophyll content)." Here, the statement correctly interprets the p-value as indicating significance at the 0.05 level. It aligns with the t-test result for OC3 (t = 7.193, p = 4.34e-07) and with the overall F-test for the model, which also indicates significance. The interpretation is consistent with standard regression inference: a p-value far below 0.05 means the predictor contributes significantly to explaining variation in HPLC.
Option C: "As P < 0.05, we fail to reject the null hypothesis and conclude OC3 (Ocean Colour) is not a significant predictor of HPLC (Chlorophyll content)." This is a direct contradiction of the reported p-value. If P < 0.05, we reject the null hypothesis of no effect, not fail to reject it. The correct inference would be that OC3 is a significant predictor, not insignificant. This option confuses the meaning of the p-value and the decision rule.
Option D: "As we have 21 degrees of freedom we can accept the null hypothesis and conclude ocean colour is not a significant predictor of chlorophyll content." Degrees of freedom alone do not determine significance; what matters is the p-value associated with the test of the predictor (and the F-statistic). Here, the p-value is extremely small, indicating rejection of the null hypothesis of no effect. Therefore this option misinterprets both the p-value logic and the significance conclusion, making it incorrect.
In summary, Option B aligns with the standard interpretation of the regression output: a p-value well below 0.05 for OC3 indicates a statistically significant predictor, supported by the large t-value and the strong F-statistic. Options A, C, and D misinterpret the p-values, significance tests, or rely on incorrect reasoning about degrees of freedom and hypothesis testing.

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