Question

Goodness of Fit

Microhabitat factors associated with forage and bed sites of barking deer in Hainan Island, China, were examined. The sample of 477 examined sites where the deer forage were categorized by habitat as follows:

Habitat	Woods	Cultivated grassplots	Deciduous forests	Other
Deer forage sites	15	16	50	386

In this region, woods make up 4.8% of the land, cultivated grassplots make up 14.7%, and deciduous forests make up 39.6%.
Do these data provide convincing evidence that barking deer prefer to forage in certain habitats over others? Answer this research question by conducting a goodness of fit test at 0.01 significance level. Follow the steps outlined in the assignment instructions.

1. Create a vector of assumed probabilities and use it to obtain and display a vector of expected values, then check that the assumptions required for this test are satisfied.

2. Find the P-value of the statistic using the appropriate chi-squared distribution.

3. Plot the distribution, a vertical line at the value of the test statistic, and show the P-value in the plot.

4. Verify that your statistic and P-value are correct by using the chisq.test instruction (if less than 10^ (-10),

   consider both as zero).

5. (h) Do these data provide evidence that barking deer prefer to forage in certain habitats over others? Explain.

THIS MUST ALL BE COMPLETED AND DISPLAYED USING R STUDIO

Answer 1

observed <- c(12,22,66,392)

p <- c(.048,.147,.396,1-(.048+.147+.396))
expected <- sum(observed) * p

expected

y <- (expected -observed)^2/ expected
TS <- sum(y)
TS
p_val <- 1-pchisq(TS,3)
p_val

chisq.test(observed,p=p)

running code

observed <- c(12,22,66,392)
> 
> p <-  c(.048,.147,.396,1-(.048+.147+.396))
> expected <- sum(observed) * p
> expected
[1]  23.616  72.324 194.832 201.228
> y <- (expected -observed)^2/ expected
> y
[1]   5.713561  35.016108  85.189724 180.859304
> TS <- sum(y)
> sum(TS)
[1] 306.7787
> p_val <- 1-pchisq(TS,3)
> p_val
[1] 0

 chisq.test(observed,p=p)

        Chi-squared test for given probabilities

data:  observed
X-squared = 306.78, df = 3, p-value < 2.2e-16

c)
(E_i) = n * p_i

Observed Expected Category Num Woods Cultivated grassplots Deciduous forests Others Sum Num 15 22.416 2.45 3473233 16 68.64940.37811477 50 386 467 84.93298.45048247 191.003199.0745172 340.3565877 467

d)

All Expected value should be greater than 5, hence the the assumptions required for this test are satisfied.

e)

\chi ^{2}=\sum \left [\frac{(O_{r,c}-E_{r,c})^{2}}{E_{r,c}} \right ]
X² = 340.37

where DF is the degrees of freedom, k is the number of levels of the categorical variable, n is the number of observations in the sample, E_i is the expected frequency count for level i, O_i is the observed frequency count for level i, and X² is the chi-square test statistic.

The P-value is the probability that a chi-square statistic having 3 degrees of freedom is more extreme than 340.37.

1.225 3.674 6.124 8.573 11.12 13.47 5.92 x2-340.3

We use the Chi-Square Distribution Calculator to find P(X² > 340.37) = less than 0.001.

Interpret results. Since the P-value (almost 0) is less than the significance level (0.05), we cannot accept the null hypothesis.

h) We have sufficient evidence in the favor of the claim that Barking deer prefer to forage in certain habitats over others.