Question

you are conducting a multinomial hypothesis test for the claim that the 4 categories occur with the following frequencies:
$h_{o}: p_{a}=0.1 ; p_{b}=0.25 ; p_{c}=0.25 ; p_{d}=0.4$
complete the table.

what is the chi - square test - statistic for this data?
$x^{2}=$
report all answers accurate to three decimal places.
very important note: answers are reported to 3 decimal places, but you should use all the decimal places to calculate the final chi - square test statistic...do not use the rounded number you report in the table.
question help: message instructor

Explanation:

Step1: Calculate total observed frequency

Total observed = 9 + 22 + 14 + 31 = 76

Step2: Compute expected frequencies

Expected = Total × probability
A: \(76×0.1 = 7.6\)
B: \(76×0.25 = 19\)
C: \(76×0.25 = 19\)
D: \(76×0.4 = 30.4\)

Step3: Calculate squared residuals

Squared residual = \(\frac{(Observed - Expected)^2}{Expected}\)
A: \(\frac{(9 - 7.6)^2}{7.6} = \frac{1.96}{7.6} ≈ 0.2579\)
B: \(\frac{(22 - 19)^2}{19} = \frac{9}{19} ≈ 0.4737\)
C: \(\frac{(14 - 19)^2}{19} = \frac{25}{19} ≈ 1.3158\)
D: \(\frac{(31 - 30.4)^2}{30.4} = \frac{0.36}{30.4} ≈ 0.0118\)

Step4: Sum squared residuals for chi-square

\(\chi^2 = 0.2579 + 0.4737 + 1.3158 + 0.0118 ≈ 2.059\)

Answer:

Completed table:

Category	Observed Frequency	Expected Frequency	Squared Residual
B	22	19	0.474
C	14	19	1.316
D	31	30.4	0.012

Chi-square test statistic: 2.059