Question

In large sample, the critical value for the single tailed test at 5% level of significance is?

• A. $2.58$
• B. $1.96$
• C. $1.645$
• D. $1.28$

Hint:

• For a single-tailed test at $\alpha = 0.05$, we need $Z_{0.05}$ such that $P(Z>Z_{0.05}) = 0.05$ (or $P(Z<-Z_{0.05}) = 0.05$).
• This means the cumulative probability $\Phi(Z_{0.05}) = 0.95$.
• Standard Z-values:
  • $Z_{0.05} \approx 1.645$ (for 5\% significance, one-tailed)
  • $Z_{0.025} \approx 1.96$ (for 2.5\% significance, one-tailed; or 5\% two-tailed)
  • $Z_{0.01} \approx 2.326$ (for 1\% significance, one-tailed)
  • $Z_{0.005} \approx 2.576$ (for 0.5\% significance, one-tailed; or 1\% two-tailed, often $2.58$)
• The option $1.642$ is the closest to $1.645$.

Answer 1

The Correct Option is C

In hypothesis testing, the critical value for a single-tailed test at a 5% level of significance is determined by the standard normal distribution (Z-distribution). For a one-tailed test, we look for the Z-value that corresponds to the area in the tail beyond 5% of the distribution.

For a one-tailed test at the 5% level of significance, the critical value (Z-critical) is:

\[ Z_{\alpha} = 1.645 \]

This means that for a one-tailed test, if the computed test statistic exceeds 1.645, we reject the null hypothesis. If the computed test statistic is less than 1.645, we fail to reject the null hypothesis.

Thus, the critical value for a single-tailed test at the 5% level of significance is 1.645.