**Critical values: $z 0.005=2.575$,$z 0.01=2.325$, $z 0.025=1.96$, $z 0.05=1.645$, $z 0.1=1.282$ when $d.f=31:t 0.005=2.744$,$t 0.01=2.453$,$t0.025=2.040$,$t0.05=1.696$,$t0.1=1.309$.**

This **article aims** to find that **less than one-quarter of soldiers** were Royalists given** significant value. **A **critical value** is a **cutoff value** used to mark the beginning of the region within which the test statistic obtained in hypothesis testing is unlikely to fall. In** hypothesis testing**, critical value is compared with test statistic obtained to determine whether or not the **null hypothesis** must be **rejected. **The critical value **divides the graph into acceptance and rejection region**s for hypothesis testing.

A **critical value **is a value that is compared to a test statistic in hypothesis testing to determine whether null hypothesis should be rejected or not. If value of the **test statistic is less extreme than the critical value**, the null hypothesis cannot be rejected. However, if the **test statistic** is more powerful than the critical value, **null hypothesis is rejected**, and alternative hypothesis is accepted. In other words, **critical value divides the distribution plot into acceptance and rejection regions**. If the value of the test statistic falls within the rejection region, then the **null hypothesis is rejected**. Otherwise, it cannot be dismissed.

Depending on the **type of distribution** to which the test statistic belongs, there are different formulas for calculating the critical value. A **confidence interval** or significance level can determine the **critical value.**

**Expert Answer**

**Step 1**

It is given that:

\[X-226\]

\[n-774\]

**Sample projection:**

\[\hat{p}-\dfrac{x}{n}=\dfrac{226}{774}=0.292\]

The** researcher claims** that **less than a quarter** of the soldiers were Royalists.

Thus,** null and alternative hypotheses** are:

\[H_{0}=p-0.25\]

\[H_{1}=p<0.25\]

**Step 2**

The **standardized test statistic** can be found as:

\[Z=\dfrac{\hat{p}-p}{\sqrt{\dfrac{p(1-p)}{n}}}\]

\[Z=\dfrac{0.292-0.25}{\sqrt{\dfrac{0.25(1-0.25)}{1200}}}=2.698\]

The **level of significance**, $=0.05$

Using $z-table$, the **critical value at the level of significance** $0.05$ is $-1.645$.

Since** calculated statistic** value $Z=2.698>|critical\:value|=|-1.645|$ ,We reject the null hypothesis. Therefore, it was **concluded** that** less than one-quarter** of the soldiers were **royalists.**

**Numerical Result**

Since** calculated statistic** value $Z=2.698>|critical\:value|=|-1.645|$ , we reject the null hypothesis. Therefore, it was **concluded** that** less than one-quarter** of the soldiers were **R****oyalists.**

**Example**

In random sample of soldiers who fought in the Battle of Preston, $784$ soldiers who fought in the Battle of Preston, $784$ soldiers were from New Model Army, $226$ were from New Model Army, and $226$ were from Royalist Army. Use the $0.1$ significance level to test the claim that less than one-quarter of the soldiers were royalists.

**Critical values are given by : $z 0.005=2.575$,$z 0.01=2.325$, $z 0.025=1.96$, $z 0.05=1.645$, $z 0.1=1.282$ when $d.f=31:t 0.005=2.744$,$t 0.01=2.453$,$t 0.025=2.040$,$t 0.05=1.696$,$t 0.1=1.309$.**

**Solution**

**Step 1**

It is given that:

\[X-226\]

\[n-784\]

**Sample projection:**

\[\hat{p}-\dfrac{x}{n}=\dfrac{226}{784}=0.288\]

The** researcher claims** that **less than a quarter** of the soldiers were Royalists.

Thus,** null and alternative hypotheses** are:

\[H_{0}=p-0.25\]

\[H_{1}=p<0.25\]

**Step 2**

The **standardized test statistic** can be found as:

\[Z=\dfrac{\hat{p}-p}{\sqrt{\dfrac{p(1-p)}{n}}}\]

\[Z=\dfrac{0.288-0.25}{\sqrt{\dfrac{0.25(1-0.25)}{1200}}}=3.04\]

The **level of significance**, $=0.1$

Using $z-table$, the **critical value at the level of significance** $0.1$ is $-1.282$.

Since** calculated statistic** $Z=3.04>|critical\:value|=|-1.282|$ , we reject the null hypothesis. Therefore, it was **concluded** that** less than one-quarter** of the soldiers were **R****oyalists.**