Minke Whales. Entanglement of marine mammals in fishing gear is a global issue and a significant threat to minke whales, in particular. In the article "Fishing gears involved in Entanglements of Minke Whales (Balaenoptera acutorostrata) in the East Sea of Korea" (Marine Mammal Science, Vol. $26$, No. $2$, Pp, $282-295$), K Song et al, studied the body lengths of minke whales entangled near Korea. The following table provides summary statistics for the body lengths of minke whales entangled at different ocean depths. Both variables, body length and ocean depth are in meters.

At the $1\%$significance level, do the data provide sufficient evidence to conclude that a difference exists in mean body length among minke whales entangled at the four different depths? Note: For the degrees of freedom in this exercise:

The given data is

The level of significance $\alpha =0.10$

Let us see the below test hypothesis

Null hypothesis

$H_0$: There is no difference exist in mean body length among minke whales entangled at the four different depths

Alternative hypothesis

$H_a$: There is a difference exist in mean body length among minke whales entangled at the four different depths

The mean of all observations is

$\overline{x}=\frac{39(4.49)+20(5.06)+28(5.75)+14(5.99)}{39+20+28+14}$

$=5.160$

The treatment sum of squares is

$SSTR=\sum n_i{\left({\overline{x}}_i-\overline{x}\right)}^2$

$=39(4.49-5.160{)}^2+20(5.06-5.160{)}^2+28(5.75-5.160{)}^2$

$+14(5.99-5.160{)}^2$

$=37.099$

The error sum of squares is

$SSE=\sum \left(n_i-1\right){\left(s_i\right)}^2$

$=(39-1)(0.80{)}^2+(20-1\left)\right(0.79{)}^2+(28-1)(1.20{)}^2$

$+(14-1)(0.97{)}^2$

$=87.290$

The total sum of squares is

$SST=SSTR+SSE$

$=37.099+87.290$

$=124.389$

The mean treatment of sum of squares is

$MSTR=\frac{SSTR}{k-1}$

$=\frac{37.099}{4-1}$

$=12.36$

The mean error of the sum of squares is

$MSE=\frac{SSE}{n-k}$

$=\frac{89.290}{101-4}$

$=0.900$

The $F-$static is

$F-static=\frac{MSTR}{MSE}$

$=\frac{12.366}{0.900}$

$=13.74$

From the given data

$F-$static $(13.74)>$critical value $(3.99)$

As a result, the crucial value approach

At a $1\%$significant threshold, the null hypothesis is rejected.

As a result, the results are statistically significant at the$1\%$level.

As a result, the results show that there is a variation in mean body length across minke whales entangled at four different depths.