Find the two-tailed p-value for a z statistic of 2.1.

Step-by-step solution

1. 1

   Say what a p-value measures. It is the probability, assuming the null hypothesis is true, of getting a test statistic at least as extreme as the one observed. For a two-tailed test, 'extreme' means far from zero in either direction.

2. 2

   Find the upper-tail area. From the standard normal distribution,

   $P(Z>2.1)=1-\Phi(2.1)=1-0.982136=0.017864.$

   A table lookup gives $\Phi(2.1)=0.9821$; the exact value from the error function is $0.9821356$.

3. 3

   Double it for the second tail. By symmetry $P(Z<-2.1)$ equals the same $0.017864$, so

   $p=2\times 0.017864=0.035729\approx 0.0357.$

   Forgetting to double is the single most common error here — it would make the result look twice as significant as it is.

4. 4

   Compare with the usual thresholds. Since $0.0357<0.05$ the result is significant at the $5\%$ level, but since $0.0357>0.01$ it is not significant at the $1\%$ level. Equivalently, $2.1$ exceeds the two-tailed critical value $1.96$ but falls short of $2.576$.

5. 5

   State the conclusion carefully. Reject $H_0$ at $\alpha=0.05$. The $p$-value is the probability of data this extreme given $H_0$ — not the probability that $H_0$ is true, which is a different and much-misused statement.