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Find the critical value z a/2 that corresponds to a 93% confidence level.

This question belongs to the statistics domain and aims to understand the alpha level, confidence level, z-critical values, the term $z_{\alpha /2}$ and further explains how to calculate these parameters.

The alpha level or significance level is the probability of producing a false decision when the null hypothesis is correct. Alpha levels are employed in hypothesis tests. Commonly, these trials are conducted with an alpha level of $0.05$ $(5\%)$, but other levels typically used are $.01$ and $.10$. Alpha levels are connected to confidence levels. To get $\alpha$, subtract the confidence level from $1$. For example, if you desire to be $95$ percent confident that your research is accurate, the alpha level would be $1-0.95$ = $5$ percent, supposing you had a one-tailed trial. For two-tailed trials, divide the alpha level by $2$. In this instance, the two-tailed alpha would be $\dfrac{0.05}{2} = 2.5\%$.

The confidence coefficient is the confidence level declared as a proportion, instead of a percentage. For instance, if your confidence level is $99\%$, the confidence coefficient would be $.99$. In broad, the greater the coefficient, the more confident you are that your results are precise. For instance, a $.99$ coefficient is more precise than a coefficient of $.89$. It’s quite rare to see a coefficient of $1$ (meaning that you are true without a suspicion that your results are complete $100\%$ authentic). A coefficient of $0$ indicates that you have no confidence that your results are factual at all.

Whenever you come across the phrase $z_{\alpha /2}$ in statistics, it is entirely directed to the z critical value from the z table that approximates $\dfrac{\alpha}{2}$.

Consider we want to see  $z_{\alpha /2}$ for some trial that is utilizing a $90%$ confidence level.

In this scenario, $\alpha$ would be $1–0.9$ = $0.1$. Thus, $\dfrac{\alpha}{2}$ = $\dfrac{0.1}{2}$ = $0.05$.

To calculate the connected z critical value, we would just look for $0.05$ in a z table. Notice that the actual value of $0.05$ doesn’t arise in the table, but it would be straight between the numbers $.0505$ and $.0495$. The related z-critical values on the exterior of the table are $-1.64$ and $-1.65$.

By dividing the difference, we notice that the z-critical value would be $-1.645$. And generally, when we utilize $z_{\alpha /2}$ we obtain the absolute value. Consequently, $z_{0.1/2}$ = $1.645$.

Expert Answer

Confidence Level is given as $C.L \space = \space 93\%$

Confidence coefficient is $0.93$

Alpha $\alpha$ comes out to be:

\[ \alpha = \space 1 – 0.93 \]

\[ \alpha = \space 0.07  \]

Calculating $\alpha /2$ by dividing both sides by $2$.

\[ \dfrac{\alpha}{2} = \space \dfrac{0.07}{2}  \]

\[ \dfrac{\alpha}{2} = \space 0.035  \]

Finding $z$ such that $P(Z>z)= 0.035$

\[= P(Z<z) =1-0.035 =0.965\]

$z$ comes out to be:

\[z = 1.81\]

Numerical Result

The critical value $z_{\alpha/2}$ that corresponds to a $93 \%$ confidence level is $1.81$.

Example

Find $z_{\alpha/2}$ for $98\%$ confidence.

\[ \alpha=1-0.98 \]

\[\alpha=0.02\]

\[\dfrac{\alpha}{2}=\dfrac{0.02}{2}\]

\[ \dfrac{\alpha}{2} =0.01\]

From the z-table, it can be seen that $z_{0.01}$ is $2.326$.

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