MHB Can we determine the preference of the pilfering opossums?

[anemone]

Problem:

A farmer plants 3 types of blueberries, namely highbush, lowbush and hybrid half-high in the ratio 5:3:2. The yield of large blueberries among the types are 30% for highbush, 40% for lowbush and 60% for hybrid half-high. It is found that opossums eat 5% of highbush blueberries, 10% of lowbush blueberries and 20% of hybrid half-high blueberries. Is there any evidence to show that opossums prefer small blueberries?

Attempt:
First, I find the percentage of the large blueberries are eaten by the opossums as follows:
P(large blueberries are eaten by the opossums)

=P(L | E)

$\displaystyle =\frac{(0.5 \times 0.3 \times 0.05)+(0.3 \times 0.4 \times 0.1)+(0.2 \times 0.6 \times 0.2)}{(0.5 \times 0.05)+(0.3 \times 0.1)+(0.2 \times 0.2)} \times 100\%$

$\displaystyle =\frac{0.0435}{0.095} \times 100\%$

$\displaystyle =45.8\%$


Second, I get the percentage of the small blueberries that are eaten by the opossums as 100%-45.8%=54.2%.

Therefore, we can say that there is enough evidence to show that opossums prefer small blueberries because the percentage that we obtained for the small blueberries that are eaten by the opossums is 54.2%, which is more than 50%.

Can someone please tell me is my working valid?

Thanks in advance.(Smile)

 

[I like Serena]

Problem:

A farmer plants 3 types of blueberries, namely highbush, lowbush and hybrid half-high in the ratio 5:3:2. The yield of large blueberries among the types are 30% for highbush, 40% for lowbush and 60% for hybrid half-high. It is found that opossums eat 5% of highbush blueberries, 10% of lowbush blueberries and 20% of hybrid half-high blueberries. Is there any evidence to show that opossums prefer small blueberries?

Attempt:
First, I find the percentage of the large blueberries are eaten by the opossums as follows:
P(large blueberries are eaten by the opossums)

=P(L | E)

$\displaystyle =\frac{(0.5 \times 0.3 \times 0.05)+(0.3 \times 0.4 \times 0.1)+(0.2 \times 0.6 \times 0.2)}{(0.5 \times 0.05)+(0.3 \times 0.1)+(0.2 \times 0.2)} \times 100\%$

$\displaystyle =\frac{0.0435}{0.095} \times 100\%$

$\displaystyle =45.8\%$


Second, I get the percentage of the small blueberries that are eaten by the opossums as 100%-45.8%=54.2%.

Therefore, we can say that there is enough evidence to show that opossums prefer small blueberries because the percentage that we obtained for the small blueberries that are eaten by the opossums is 54.2%, which is more than 50%.

Can someone please tell me is my working valid?

Thanks in advance.

Hi anemone! :)

What do you mean by the symbol E?
Anyway, you've found that the opossums eat 45.8% large and 54.2% small for a total of 100%.
But... what happened to the hybrid blueberries?Let's start with how much of each plant we get:

That would be a total of 5x30 + 3x40 + 2x60 = 390 plants (with an arbitrary factor that we'll set to 1 without loss of generality).
The amount of highbush is 5x30 = 150.
The amount of lowbush is 3x40 = 120.
The amount of hybrid is 2x60 = 120.

If the opossums didn't care, they would likely eat blueberries in this ratio (null hypothesis H0).

The total that we have observed the opossums to eat is 5% x 150 + 10% x 120 + 20% x 120 = 43.5 plants.
They eat 5% large, which a corresponding fraction of 5% x 150 / (5% x 150 + 10% x 120 + 20% x 120) = 17%
They eat 10% low for 10% x 120 / (5% x 150 + 10% x 120 + 20% x 120) = 28%
They eat 20% hybrid for 20% x 120 / (5% x 150 + 10% x 120 + 20% x 120) = 55%.
Checking... yes the total is 100%.

What we see is that the opossums prefer hybrid by far.
Small blueberries are their second choice.

 

[anemone]

Hi anemone! :)

What do you mean by the symbol E?
Anyway, you've found that the opossums eat 45.8% large and 54.2% small for a total of 100%.
But... what happened to the hybrid blueberries?

By the symbol E, I meant the blueberries (all 3 types of them) that are eaten by opossums...

Let's start with how much of each plant we get:

That would be a total of 5x30 + 3x40 + 2x60 = 390 plants (with an arbitrary factor that we'll set to 1 without loss of generality).
The amount of highbush is 5x30 = 150.
The amount of lowbush is 3x40 = 120.
The amount of hybrid is 2x60 = 120.

If the opossums didn't care, they would likely eat blueberries in this ratio (null hypothesis H0).

The total that we have observed the opossums to eat is 5% x 150 + 10% x 120 + 20% x 120 = 43.5 plants.
They eat 5% large, which a corresponding fraction of 5% x 150 / (5% x 150 + 10% x 120 + 20% x 120) = 17%
They eat 10% low for 10% x 120 / (5% x 150 + 10% x 120 + 20% x 120) = 28%
They eat 20% hybrid for 20% x 120 / (5% x 150 + 10% x 120 + 20% x 120) = 55%.
Checking... yes the total is 100%.

What we see is that the opossums prefer hybrid by far.
Small blueberries are their second choice.

Thanks for answering to my post, ILikeSerena! (Smile)

But I need some time to digest the explanation above. (Tmi)

 

[I like Serena]

By the symbol E, I meant the blueberries (all 3 types of them) that are eaten by opossums...

Ah, I see what you mean now.

I'd say that the proportion of large blueberries that opossums eat is:

$P(L | E) = \dfrac{P(L \wedge E)}{P(E)} = \dfrac{\text{fraction of large blueberries eaten}}{\text{total fraction eaten}}$

$P(L | E) = \dfrac{0.5 \times 0.3 \times 0.05}{(0.5 \times 0.3 \times 0.05)+(0.3 \times 0.4 \times 0.1)+(0.2 \times 0.6 \times 0.2)}
\times 100\%= 17.2\%$

$P(S | E) = \dfrac{0.3 \times 0.4 \times 0.1}{(0.5 \times 0.3 \times 0.05)+(0.3 \times 0.4 \times 0.1)+(0.2 \times 0.6 \times 0.2)} \times 100\%= 27.6\%$

$P(H | E) = \dfrac{0.2 \times 0.6 \times 0.2}{(0.5 \times 0.3 \times 0.05)+(0.3 \times 0.4 \times 0.1)+(0.2 \times 0.6 \times 0.2)} \times 100\%= 55.2\%$