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In a random sample of soldiers who fought in the Battle of Preston, 774 soldiers were from the New Model Army, and 226 were from the Royalist Army. Use a 0.05 significance level to test the claim that fewer than one quarter of the soldiers were Royalist.

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 regions 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 Royalists.

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 Royalists.

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