A proofreader checked 160 ads for grammatical errors. The sample frequency distribution is shown: $$\begin{array} { | l | c | c | c | c | }
\hline \text { Number of Errors } & 0 & 1 & 2 & 3 \
\hline \text { Observed Frequency } & 10 & 65 & 71 & 14 \
\hline
\end{array}$$ Using a goodness-of-fit test to determine whether this distribution is uniform would result in a chi-square test statistic of approximately:
A)55.
B)79.
C)85.
D) 161.

Question

A proofreader checked 160 ads for grammatical errors. The sample frequency distribution is shown: $$\begin{array} { | l | c | c | c | c | } 
\hline \text { Number of Errors } & 0 & 1 & 2 & 3 \
\hline \text { Observed Frequency } & 10 & 65 & 71 & 14 \
\hline
\end{array}$$ Using a goodness-of-fit test to determine whether this distribution is uniform would result in a chi-square test statistic of approximately:
A)55.
B)79.
C)85.
D) 161.

Quizplus · Accepted Answer

To determine whether the sample frequency distribution is uniform, we need to calculate the expected frequency of each category assuming a uniform distribution:

- Expected Frequency for 0 errors: (1/4)*160 = 40
- Expected Frequency for 1 error: (1/4)*160 = 40
- Expected Frequency for 2 errors: (1/4)*160 = 40
- Expected Frequency for 3 errors: (1/4)*160 = 40

We can then use the formula for the chi-square goodness-of-fit test:

χ² = Σ (Observed Frequency - Expected Frequency)² / Expected Frequency

Plugging in the values from the frequency chart and the expected frequencies, we get:

χ² = (10 - 40)²/40 + (65 - 40)²/40 + (71 - 40)²/40 + (14 - 40)²/40
    = 30 + 6.25 + 9.225 + 6.25
    = 51.725

Degrees of freedom (df) = number of categories - 1 = 4 - 1 = 3

Using a chi-square distribution table with 3 degrees of freedom and a significance level of 0.05, the critical value is approximately 7.815. Since our calculated χ² value of 51.725 is greater than the critical value, we can reject the null hypothesis that the sample frequency distribution is uniform.

Therefore, the chi-square test statistic is 51.725, which is closest to answer choice B (79).

A Proofreader Checked 160 Ads for Grammatical Errors Using a Goodness-Of-Fit Test to Determine Whether This Distribution Is