Question

researchers studied the behavior of birds that were searching for seeds and insects in an oregon forest. in this forest, 54% of the trees were douglas firs, 40% were ponderosa pines, and 6% were other types of trees. at a randomly selected time during the day, the researchers observed 156 red-breasted nuthatches: 70 were seen in douglas firs, 79 in ponderosa pines, and 7 in other types of trees. Do these data provide convincing evidence that nuthatches prefer particular types of trees when they're searching for seeds and insects?

Answer 1

Yes, the given data provide convincing evidence that the Nuthatches prefer particular types of trees while searching for seeds and insects.

A statistical hypothesis test used to examine if a variable is likely to come from a given distribution is the Chi-square goodness of fit test. It is frequently used to assess how well sampling data represents the entire population.

Given that the number of red-breasted nuthatches (n) is 156. By convention, the significance level is α=0.05.

First, let's state the null hypothesis and alternate hypothesis.

Null hypothesis H₀: Nuthatches don't prefer particular types of trees while searching for seeds and insects.

Alternate hypothesis H₁: Nuthatches prefer particular types of trees while searching for seeds and insects.

Given: p₁=54%=0.54, p₂=40%=0.40, p₃=6%=0.06

The expected frequency (E) for each probability is:

$\begin{aligned}E_1&=np_{1}\\&=156\times0.54\\&=84.24\\E_2&=np_{2}\\&=156\times0.40\\&=62.4\\E_3&=np_{3}\\&=156\times0.06\\&=9.36\end{aligned}$

The observed values (O) are given as 70, 79, and 7. Calculating chi-square, we get,

$\begin{aligned}\chi^{2}&=\sum\frac{(O-E)^2}{E}\\&=\frac{(70-84.24)^2}{84.24}+\frac{(79-62.4)^2}{62.4}+\frac{(7-9.36)^2}{9.36}\\&=7.4182\end{aligned}$

And the degree of freedom is given by df = n-1, where, n is the sample size. Here, the sample size is 3.

$\begin{aligned}df&=n-1\\&=3-1\\&=2\end{aligned}$

Then, the P-value from the table is given by, $P(\chi^{2} > 7.4182) = 0.0245$.

This value is less than the significance level.  Therefore, the null hypothesis is rejected. So, an alternate hypothesis is accepted.

Therefore,  nuthatches prefer particular types of trees while searching for seeds and insects.

To know more about the chi-square test:

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