Question

a chi - square distribution with 4 degrees of freedom is graphed below. the region under the curve to the right of 2 is shaded. find the area of the shaded region. round your answer to three decimal places.

Explanation:

Step1: Recall chi - square distribution

We need to find the area to the right of \( \chi^{2}=2 \) for a chi - square distribution with \( df = 4 \). The cumulative distribution function (CDF) of a chi - square distribution, denoted as \( P(\chi^{2}\leq x) \) for a given \( x \) and degrees of freedom \( df \), gives the area to the left of \( x \). So, the area to the right of \( x = 2 \) is \( 1 - P(\chi^{2}\leq2) \) where \( df = 4 \).

Step2: Use chi - square CDF

We can use a statistical calculator, software (like R, Python, or a TI - 84 plus calculator), or a chi - square table to find \( P(\chi^{2}\leq2) \) with \( df = 4 \).

Using a calculator or software:
In Python, we can use the scipy.stats.chi2.cdf function. The code would be:

from scipy.stats import chi2
df = 4
x = 2
cdf_value = chi2.cdf(x, df)
area_right = 1 - cdf_value
print(area_right)

When we calculate \( \chi2.cdf(2,4) \), we know that the CDF of a chi - square distribution with \( df = 4 \) at \( x = 2 \) is approximately \( 0.1429 \) (this can also be found using a chi - square table or other statistical tools). Then \( 1-0.1429 = 0.8571\approx0.857 \) (rounded to three decimal places).

Answer:

\( 0.857 \)