Binary outcomes

• $c={\arg \text{min}}_{k>N{\pi }_0}\left[P_{\text{Bi}}\left(X\ge k;N,{\pi }_0\right)\le \alpha \right]$
• $P_{\text{Bi}}\left(X\ge c;N,{\pi }_1\right)$

• Correctly identified the two steps of the solution.
• Correctly determined the critical value from the upper tail of the binomial distribution.
• Correctly calculated the critical value from the cumulative distribution.
• Correctly determined the power from the upper tail of the distribution.

• $c=\underset{\text{instead of}{\arg \text{min}}_{k>N{\pi }_0}}{\underbrace{{\arg \text{max}}_{k<N{\pi }_0}}}\left[P_{\text{Bi}}\left(\underset{\text{instead of}X\ge k}{\underbrace{X\le k}};N,{\pi }_0\right)\le \alpha \right]$
• $P_{\text{Bi}}\left(\underset{\text{instead of}X\ge c}{\underbrace{X=c}};N,{\pi }_1\right)$

• Correctly identified the two steps of the solution.
• The result matches the lower critical value of the binomial distribution. Please use the upper tail to determine the critical value.
• Correctly calculated the critical value from the cumulative distribution.
• The result matches the power based on the binomial probability, $P_{\text{Bi}}\left(X=c\right)$. Please determine the power based on the cumulative distribution, $P_{\text{Bi}}\left(X\ge c\right)$.

• $c=\underset{\text{instead of}{\arg \text{min}}_{k>N{\pi }_0}}{\underbrace{{\arg \text{max}}_{k<N{\pi }_0}}}\left[P_{\text{Bi}}\left(\underset{\text{instead of}X\ge k}{\underbrace{X\le k}};N,{\pi }_0\right)\le \alpha \right]$
• $P_{\text{Bi}}\left(\underset{\text{instead of}X\ge c}{\underbrace{X\le c}};N,{\pi }_1\right)$

• Correctly identified the two steps of the solution.
• The result matches the lower critical value of the binomial distribution. Please use the upper tail to determine the critical value.
• Correctly calculated the critical value from the cumulative distribution.
• The result matches the power based on the lower tail of the binomial distribution. Please use the upper tail to calculate the power.

• $c=\underset{\text{instead of}{\arg \text{min}}_{k>N{\pi }_0}}{\underbrace{{\arg \text{max}}_{k<N{\pi }_0}}}\left[P_{\text{Bi}}\left(\underset{\text{instead of}X\ge k}{\underbrace{X=k}};N,{\pi }_0\right)\le \alpha \right]$
• $P_{\text{Bi}}\left(\underset{\text{instead of}X\ge c}{\underbrace{X=c}};N,{\pi }_1\right)$

• Correctly identified the two steps of the solution.
• The result matches the lower critical value of the binomial distribution. Please use the upper tail to determine the critical value.
• The result matches the critical value based on the binomial probability, $P_{\text{Bi}}\left(X=k\right)$. Please calculate the critical value based on the cumulative distribution, $P_{\text{Bi}}\left(X\ge k\right)$.
• The result matches the power based on the binomial probability, $P_{\text{Bi}}\left(X=c\right)$. Please determine the power based on the cumulative distribution, $P_{\text{Bi}}\left(X\ge c\right)$.

• $c=\underset{\text{instead of}{\arg \text{min}}_{k>N{\pi }_0}}{\underbrace{{\arg \text{max}}_{k<N{\pi }_0}}}\left[P_{\text{Bi}}\left(\underset{\text{instead of}X\ge k}{\underbrace{X=k}};N,{\pi }_0\right)\le \alpha \right]$
• $P_{\text{Bi}}\left(\underset{\text{instead of}X\ge c}{\underbrace{X\le c}};N,{\pi }_1\right)$

• Correctly identified the two steps of the solution.
• The result matches the lower critical value of the binomial distribution. Please use the upper tail to determine the critical value.
• The result matches the critical value based on the binomial probability, $P_{\text{Bi}}\left(X=k\right)$. Please calculate the critical value based on the cumulative distribution, $P_{\text{Bi}}\left(X\ge k\right)$.
• The result matches the power based on the lower tail of the binomial distribution. Please use the upper tail to calculate the power.

• $c={\arg \text{min}}_{k>N{\pi }_0}\left[P_{\text{Bi}}\left(\underset{\text{instead of}X\ge k}{\underbrace{X=k}};N,{\pi }_0\right)\le \alpha \right]$
• $P_{\text{Bi}}\left(\underset{\text{instead of}X\ge c}{\underbrace{X=c}};N,{\pi }_1\right)$

• Correctly identified the two steps of the solution.
• Correctly determined the critical value from the upper tail of the binomial distribution.
• The result matches the critical value based on the binomial probability, $P_{\text{Bi}}\left(X=k\right)$. Please calculate the critical value based on the cumulative distribution, $P_{\text{Bi}}\left(X\ge k\right)$.
• The result matches the power based on the binomial probability, $P_{\text{Bi}}\left(X=c\right)$. Please determine the power based on the cumulative distribution, $P_{\text{Bi}}\left(X\ge c\right)$.

• $c={\arg \text{min}}_{k>N{\pi }_0}\left[P_{\text{Bi}}\left(X\ge k;N,{\pi }_0\right)\le \alpha \right]$
• $P_{\text{Bi}}\left(\underset{\text{instead of}X\ge c}{\underbrace{X=c}};N,{\pi }_1\right)$

• Correctly identified the two steps of the solution.
• Correctly determined the critical value from the upper tail of the binomial distribution.
• Correctly calculated the critical value from the cumulative distribution.
• The result matches the power based on the binomial probability, $P_{\text{Bi}}\left(X=c\right)$. Please determine the power based on the cumulative distribution, $P_{\text{Bi}}\left(X\ge c\right)$.

• $c={\arg \text{min}}_{k>N{\pi }_0}\left[P_{\text{Bi}}\left(\underset{\text{instead of}X\ge k}{\underbrace{X=k}};N,{\pi }_0\right)\le \alpha \right]$
• $P_{\text{Bi}}\left(\underset{\text{instead of}X\ge c}{\underbrace{X\le c}};N,{\pi }_1\right)$

• Correctly identified the two steps of the solution.
• Correctly determined the critical value from the upper tail of the binomial distribution.
• The result matches the critical value based on the binomial probability, $P_{\text{Bi}}\left(X=k\right)$. Please calculate the critical value based on the cumulative distribution, $P_{\text{Bi}}\left(X\ge k\right)$.
• The result matches the power based on the lower tail of the binomial distribution. Please use the upper tail to calculate the power.

• $c={\arg \text{min}}_{k>N{\pi }_0}\left[P_{\text{Bi}}\left(X\ge k;N,{\pi }_0\right)\le \alpha \right]$
• $P_{\text{Bi}}\left(\underset{\text{instead of}X\ge c}{\underbrace{X\le c}};N,{\pi }_1\right)$

• Correctly identified the two steps of the solution.
• Correctly determined the critical value from the upper tail of the binomial distribution.
• Correctly calculated the critical value from the cumulative distribution.
• The result matches the power based on the lower tail of the binomial distribution. Please use the upper tail to calculate the power.

• $c={\arg \text{min}}_{k>N{\pi }_0}\left[P_{\text{Bi}}\left(\underset{\text{instead of}X\ge k}{\underbrace{X=k}};N,{\pi }_0\right)\le \alpha \right]$
• $P_{\text{Bi}}\left(X\ge c;N,{\pi }_1\right)$

• Correctly identified the two steps of the solution.
• Correctly determined the critical value from the upper tail of the binomial distribution.
• The result matches the critical value based on the binomial probability, $P_{\text{Bi}}\left(X=k\right)$. Please calculate the critical value based on the cumulative distribution, $P_{\text{Bi}}\left(X\ge k\right)$.
• Correctly determined the power from the upper tail of the distribution.

• $c=\underset{\text{instead of}{\arg \text{min}}_{k>N{\pi }_0}}{\underbrace{{\arg \text{max}}_{k<N{\pi }_0}}}\left[P_{\text{Bi}}\left(\underset{\text{instead of}X\ge k}{\underbrace{X=k}};N,{\pi }_0\right)\le \alpha \right]$
• $P_{\text{Bi}}\left(X\ge c;N,{\pi }_1\right)$

• Correctly identified the two steps of the solution.
• The result matches the lower critical value of the binomial distribution. Please use the upper tail to determine the critical value.
• The result matches the critical value based on the binomial probability, $P_{\text{Bi}}\left(X=k\right)$. Please calculate the critical value based on the cumulative distribution, $P_{\text{Bi}}\left(X\ge k\right)$.
• Correctly determined the power from the upper tail of the distribution.

• $c=\underset{\text{instead of}{\arg \text{min}}_{k>N{\pi }_0}}{\underbrace{{\arg \text{max}}_{k<N{\pi }_0}}}\left[P_{\text{Bi}}\left(\underset{\text{instead of}X\ge k}{\underbrace{X\le k}};N,{\pi }_0\right)\le \alpha \right]$
• $P_{\text{Bi}}\left(X\ge c;N,{\pi }_1\right)$

• Correctly identified the two steps of the solution.
• The result matches the lower critical value of the binomial distribution. Please use the upper tail to determine the critical value.
• Correctly calculated the critical value from the cumulative distribution.
• Correctly determined the power from the upper tail of the distribution.

$P_{\text{Bi}}\left(\mathrm{tail}\left(\text{upper}\right);N,{\pi }_0\right)$

• Correctly recognized the problem as involving a binomial distribution.
• Correctly calculated the probability for $k$ or more successes under the null hypothesis.

$P_{\text{Bi}}\left(\underset{\text{instead of}\mathrm{tail}\left(\text{upper}\right)}{\underbrace{\mathrm{tail}\left(\text{densi}\right)}};N,\underset{\text{instead of}{\pi }_0}{\underbrace{{\pi }_1}}\right)$

• The result matches the probability under the alternative ${\pi }_1$.
• Correctly recognized the problem as involving a binomial distribution.
• The result matches the probability of having exactly $k$ successes.

$P_{\text{Bi}}\left(\underset{\text{instead of}\mathrm{tail}\left(\text{upper}\right)}{\underbrace{\mathrm{tail}\left(\text{densi}\right)}};N,\underset{\text{instead of}{\pi }_0}{\underbrace{{\pi }_1}}\right)$

• Correctly recognized the problem as involving a binomial distribution.
• The result matches the probability under the alternative ${\pi }_1$.
• The result matches the probability of having exactly $k$ successes.

$P_{\text{Bi}}\left(\underset{\text{instead of}\mathrm{tail}\left(\text{upper}\right)}{\underbrace{\mathrm{tail}\left(\text{densi}\right)}};N,\underset{\text{instead of}{\pi }_0}{\underbrace{{\pi }_1}}\right)$

• Correctly recognized the problem as involving a binomial distribution.
• The result matches the probability of having exactly $k$ successes.
• The result matches the probability under the alternative ${\pi }_1$.

$P_{\text{Bi}}\left(\underset{\text{instead of}\mathrm{tail}\left(\text{upper}\right)}{\underbrace{\mathrm{tail}\left(\text{lower}\right)}};N,\underset{\text{instead of}{\pi }_0}{\underbrace{{\pi }_1}}\right)$

• The result matches the probability under the alternative ${\pi }_1$.
• Correctly recognized the problem as involving a binomial distribution.
• The result matches the probability of having $k$ or less successes.

$P_{\text{Bi}}\left(\underset{\text{instead of}\mathrm{tail}\left(\text{upper}\right)}{\underbrace{\mathrm{tail}\left(\text{lower}\right)}};N,\underset{\text{instead of}{\pi }_0}{\underbrace{{\pi }_1}}\right)$

• Correctly recognized the problem as involving a binomial distribution.
• The result matches the probability under the alternative ${\pi }_1$.
• The result matches the probability of having $k$ or less successes.

$P_{\text{Bi}}\left(\underset{\text{instead of}\mathrm{tail}\left(\text{upper}\right)}{\underbrace{\mathrm{tail}\left(\text{lower}\right)}};N,\underset{\text{instead of}{\pi }_0}{\underbrace{{\pi }_1}}\right)$

• Correctly recognized the problem as involving a binomial distribution.
• The result matches the probability of having $k$ or less successes.
• The result matches the probability under the alternative ${\pi }_1$.

$P_{\text{Bi}}\left(\underset{\text{instead of}\mathrm{tail}\left(\text{upper}\right)}{\underbrace{\mathrm{tail}\left(\text{densi}\right)}};N,{\pi }_0\right)$

• Correctly recognized the problem as involving a binomial distribution.
• The result matches the probability of having exactly $k$ successes.

$P_{\text{Bi}}\left(\underset{\text{instead of}\mathrm{tail}\left(\text{upper}\right)}{\underbrace{\mathrm{tail}\left(\text{lower}\right)}};N,{\pi }_0\right)$

• Correctly recognized the problem as involving a binomial distribution.
• The result matches the probability of having $k$ or less successes.

$P_{\text{Bi}}\left(\mathrm{tail}\left(\text{upper}\right);N,\underset{\text{instead of}{\pi }_0}{\underbrace{{\pi }_1}}\right)$

• The result matches the probability under the alternative ${\pi }_1$.
• Correctly recognized the problem as involving a binomial distribution.
• Correctly calculated the probability for $k$ or more successes under the null hypothesis.

$P_{\text{Bi}}\left(\mathrm{tail}\left(\text{upper}\right);N,\underset{\text{instead of}{\pi }_0}{\underbrace{{\pi }_1}}\right)$

• Correctly recognized the problem as involving a binomial distribution.
• The result matches the probability under the alternative ${\pi }_1$.
• Correctly calculated the probability for $k$ or more successes under the null hypothesis.

$P_{\text{Bi}}\left(\mathrm{tail}\left(\text{upper}\right);N,\underset{\text{instead of}{\pi }_0}{\underbrace{{\pi }_1}}\right)$

• Correctly recognized the problem as involving a binomial distribution.
• Correctly calculated the probability for $k$ or more successes under the null hypothesis.
• The result matches the probability under the alternative ${\pi }_1$.