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 $2.326$.