MATH 105 Lab 08: Are Female Mallards Attracted to the Color Green

Of all the types of ducks, the mallard (Anas platy) is one of the most widespread and familiar species in the world. A male mallard in breeding plumage is recognizable by his iridescent green head which is bordered by a white collar. The chest is a rich chestnut and leads to grayish white underparts. The back is grayish brown and the blue speculum on the wings is bordered by thick white lines.

Are female mallards attracted to the color green?

Suppose you are a student taking a biology class that studies animal behavior and you are assigned to the following research question: In a certain species (mallards), male ducks have green heads and females are plain color. Probably the purpose of the green coloring of the male heads is to attract the females. The question is: are female ducks also attracted to the green color in food, for example bread?

Writing a statistical hypotheses. We basically want to know if female ducks are indifferent to green bread versus plain bread or if they prefer green bread.

The research question can be translated into the confrontation of two opposite ideas:

Idea 1: Female ducks are indifferent to plain versus green bread.

Idea 2: Female ducks prefer green bread.

When a female duck of the above mentioned species is confronted with two pieces of bread, one plain and one green, the probability of picking the green one will be called p. Write the two previous ideas in terms of p

We call these these confronting ideas statistical hypotheses. The first one states that the ducks equally like the green and the plain bread. This statement is called the null hypothesis because it represents an idea of no difference and is labeled by the symbol 'H0'. The second idea says that the ducks prefer the green bread and states something different than the first one, so it is called the alternative hypothesis. The symbol used for the alternative hypothesis is 'Ha'.

We must decide which of these two statistical hypotheses is more likely to be true.

Gathering Evidence to make the decision

Suppose that the student designs a study in order to be able to mkae a decision about the two statistical hypotheses. She will go to a lake near campus where mallards are quite abundant and will randomly select 10 female ducks. Each duck will be offered two pieces of bread: one plain and one dyed green. The student will write down which piece of bread each duck approaches first. Then she will summarize her information reporting how many ducks approach the green bread first.

Of course even if the null hypothesis was true we are not always going to get the results that we would expect. This is due to sampling variablilty which we learned about in Chapter 7. For instance, if we toss a fair coin 10 times we won't always get 5 heads and 5 tails. On rare occasions, we might get something like 9 heads and 1 tail.

As mentioned previously, we will need to pick a hypothesis after we examine the data. The idea with statistical hypothesis testing is that we only go with the alternative hypothesis (Idea 2) if we have convincing evidence for the alternative. It is similar to going to court where we do not convict someone unless there is evidence beyond a reasonable doubt that someone is guilty.

In hypothesis testing we assume that the null hypothesis is true and then we collect data and see how unusual our results are under that assumption.

Now that we have an idea of how to run a hypothesis test we will use to mimick what happens when we give the 10 ducks green and regular bread assuming that they have no preference. View the video below to see how to do this in StatCrunch.

https://media.csuchico.edu/media/Math+105+Lab+8+Null+Histogram/1_9ns9hdnn

The above histogram is called the null distribution.

Suppose you did this experiment and 7 of the ducks that you fed went for the green bread. Remember that for us to pick the alternative we need convincing evidence similar to what we would need in court to convict someone (evidence beyond a reasonable doubt).

The value you calculated above is called the p-value and it represents the probability of getting our sample results if the null hypothesis is true. Therefore, in context of the problem, the p-value you calculated is the probability of 7 or more ducks picking green bread even though they have no preference.