F-statistics and coefficient p-value of model with only one variable

Question

F-test tests the null hypothesis that all coefficient of variables in the model equal to zero.

P-value in a hypothesis test shows the probability of having observed results if null hypothesis is true.

My question is that if the model only has one variable, just like m2 and m3 shown as following, is the result of f-statistics just the same as the p-value?

Dataset Regarding the dataset. The original dataset has 50 states. Alaska and Texas are the outliers. From some analysis, I already decide to remove Alaska as it has a large effect on the linear model according to the regression results. Now the problem is whether Texas should be removed.

area.no.alaska has 49 states, Alaska is removed area.no.tx.al has 48 states, Alaska and Texas is removed

I am trying to compare m2 with m4 to figure out which is better.

So far, I think m2 is better because it has higher r-square, higher p-value of variable land.area even though both models have p-value very close to zero for variable land.area. I am not sure whether it is necessary to compare f-statistics as only one variable in the model and I already compared the p-value of that variable.

m2 = lm(area.no.alaska$farm.area ~ area.no.alaska$land.area)
m4 = lm(farm.area ~ land.area, data = area.no.al.tx)

summary(m4)

Call:
lm(formula = farm.area ~ land.area, data = area.no.al.tx)

Residuals:
   Min     1Q Median     3Q    Max 
-41309  -8736  -1784   4361  36054 

Coefficients:
              Estimate Std. Error t value      Pr(>|t|)    
(Intercept) -569.41436 4310.67726  -0.132         0.895    
land.area      0.46526    0.06404   7.266 0.00000000365 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 16300 on 46 degrees of freedom
Multiple R-squared:  0.5344,    Adjusted R-squared:  0.5242 
F-statistic: 52.79 on 1 and 46 DF,  p-value: 0.000000003646
```

summary(m2)

Call:
lm(formula = area.no.alaska$farm.area ~ area.no.alaska$land.area)

Residuals:
   Min     1Q Median     3Q    Max 
-50695  -8712   -228   7825  49354 

Coefficients:
                            Estimate  Std. Error t value Pr(>|t|)    
(Intercept)              -8365.76964  4367.14590  -1.916   0.0615 .  
area.no.alaska$land.area     0.62171     0.05716  10.877 1.98e-14 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 18620 on 47 degrees of freedom
Multiple R-squared:  0.7157,    Adjusted R-squared:  0.7096 
F-statistic: 118.3 on 1 and 47 DF,  p-value: 1.981e-14
```

Answer 1

The only difference between these two models is that m4 excludes one observation (Texas) that is not included in m2. That the R-squared is higher is focusing on the wrong thing. Why did you remove this observation? Generally, it is not a good idea to remove data from a model for the explicit purpose of improving fit (that is considered "cooking the data"). However, it is appropriate to remove an outlier and assess the effect it has on the analysis. It little effect, then leave it in. If it makes a big difference, then you should report both.

In the above, both models are statistically significant with a strong relationship between land area and farm area. Is there a theoretical reason for excluding Texas? Would your interpretation really be different based on m4 instead of m2?