Answered

Hypotheses for a chi-square goodness-of-fit test are given, along with the sample data and the chi-square statistic.

Hypotheses:
[tex]\[
H_0: p_A = 0.3, \; p_B = 0.3, \; p_C = 0.4
\][/tex]
[tex]\[
H_a: \text{At least one } p \text{ is not as given}
\][/tex]

Sample Data:
[tex]\[
\begin{tabular}{|c|c|c|c|}
\hline
& A & B & C \\
\hline
\text{Observed(Expected)} & 25(35) & 47(35) & 45(47) \\
\hline
\end{tabular}
\][/tex]

Chi-square statistic [tex]\(= 7.1\)[/tex]

(a) What are the degrees of freedom for the test?

(b) What is the [tex]\(p\)[/tex]-value?

Round your answer to five decimal places.

[tex]\(p\)[/tex]-value [tex]\(= \)[/tex] [tex]\(\square\)[/tex]



Answer :

GinnyAnswer
Let's go through the steps to solve the chi-square goodness-of-fit test problem:

### (a) Degrees of Freedom

The degrees of freedom (df) for a chi-square goodness-of-fit test is calculated based on the number of categories (k).

[tex]\[ df = k - 1 \][/tex]

In this problem, there are three categories: A, B, and C.

So, the calculation for the degrees of freedom is:
[tex]\[ df = 3 - 1 = 2 \][/tex]

Thus, the degrees of freedom for the test is 2.

### (b) [tex]\( p \)[/tex]-value

The [tex]\( p \)[/tex]-value is obtained by using the chi-square statistic and the degrees of freedom to determine the probability that the observed distribution is due to random chance.

Given:
- The chi-square statistic is 7.1.
- The degrees of freedom are 2.

Using these values, the [tex]\( p \)[/tex]-value can be looked up in a chi-square table or more commonly computed using statistical software.

Since we have made such calculations, the [tex]\( p \)[/tex]-value for this chi-square statistic with 2 degrees of freedom is:
[tex]\[ p\text{-value} = 0.02872 \][/tex]

### Final Answers:

(a) Degrees of freedom = 2

(b) [tex]\( p \)[/tex]-value = 0.02872

Other Questions