A psychology instructor gave a five-question true-false quiz to her class of 178 students. ) $$\begin{array} { l | c c c c c c }
\text { Number Correct } & 0 & 1 & 2 & 3 & 4 & 5 \
\hline \text { Observed } & 8 & 30 & 56 & 48 & 26 & 10
\end{array}$$ The instructor thinks that the students may have answered the questions by guessing, so that the probability that any given answer is correct is 0.5. Under the null hypothesis, the number
Of correct answers has a binomial distribution with 5 trials and success probability 0.5. Perform a chi-square test of this hypothesis. Can you reject $H _ { 0 }$ at the $\alpha = 0.01$ level?
A) Yes
B) $\mathrm { No }$

Question

A psychology instructor gave a five-question true-false quiz to her class of 178 students. ) $$\begin{array} { l | c c c c c c } 
\text { Number Correct } & 0 & 1 & 2 & 3 & 4 & 5 \
\hline \text { Observed } & 8 & 30 & 56 & 48 & 26 & 10
\end{array}$$ The instructor thinks that the students may have answered the questions by guessing, so that the probability that any given answer is correct is 0.5. Under the null hypothesis, the number
Of correct answers has a binomial distribution with 5 trials and success probability 0.5. Perform a chi-square test of this hypothesis. Can you reject $H _ { 0 }$ at the $\alpha = 0.01$ level?
A) Yes
B) $\mathrm { No }$

Quizplus · Accepted Answer

To perform a chi-square test, we first calculate the expected frequencies for each number of correct answers under the null hypothesis that students are guessing (with a probability of 0.5 for each question). The binomial distribution formula is used for this calculation: $E = n \times p^k \times (1-p)^{n-k} \times \binom{n}{k}$, where $n$ is the number of trials (5 questions), $p$ is the success probability (0.5 for guessing correctly), and $k$ is the number of successes (number of correct answers). The expected frequencies for 0 to 5 correct answers are calculated using this formula. Then, we use the chi-square formula: $\chi^2 = \sum \frac{(O-E)^2}{E}$, where $O$ is the observed frequency, and $E$ is the expected frequency. After calculating the chi-square statistic, we compare it to the critical value from the chi-square distribution table with the appropriate degrees of freedom (number of categories - 1, which is 5 in this case) and the given alpha level (0.01). Without the exact chi-square value and critical value, we cannot definitively say the result, but the process described is how one would determine whether to reject $H_0$. The answer choice is based on the assumption that after performing these calculations, the chi-square statistic does not exceed the critical value, meaning there is not enough evidence to reject the null hypothesis at the 0.01 significance level.

A Psychology Instructor Gave a Five-Question True-False Quiz to Her