Question

For Exercises 7 through \(27,\) perform these steps. a. State the hypotheses and identify the claim. b. Find the critical value(s). c. Compute the test value. d. Make the decision. e. Summarize the results. Use the traditional method of hypothesis testing unless otherwise specified. Bullying Bullying is a problem at any age but especially for students aged 12 to 18 . A study showed that \(7.2 \%\) of all students in this age bracket reported that bullied at school during the past six months with 6 th grade having the highest incidence at \(13.9 \%\) and 12 th grade the lowest at \(2.2 \%\). To see if there is a difference between public and private schools, 200 students were randomly selected from each. At the 0.05 level of significance, can a difference be concluded? $$ \begin{array}{ccc}{} & {\text { Private }} & {\text { Public }} \\ \hline \text { Sample size } & {200} & {200} \\ {\text { No. bullied }} & {13} & {16}\end{array} $$

Short answer

There is no significant difference in bullying proportions between private and public schools.

Step-by-step solution

State the Hypotheses and Identify the Claim

Let's denote the population proportion of bullied students in private schools as \( p_1 \) and in public schools as \( p_2 \). We want to test whether there is a difference between these proportions at the significance level of 0.05.
**Null Hypothesis (\( H_0 \)):** \( p_1 = p_2 \) (There is no difference between the proportions of bullied students in private and public schools.)
**Alternative Hypothesis (\( H_a \)):** \( p_1 eq p_2 \) (There is a difference between the proportions of bullied students in private and public schools.)
The claim we are testing is that there is a difference between the two proportions.

Find the Critical Values

Since we're comparing two proportions, we will use the standard normal distribution (Z-distribution) to determine the critical value for a two-tailed test at the 0.05 significance level. For a two-tailed test, the critical values correspond to the standard Z-scores:The critical values are \( Z = \pm 1.96 \). This means we'll reject the null hypothesis if our calculated Z is less than -1.96 or greater than 1.96.

Compute the Test Value

Calculate the pooled sample proportion \( p = \frac{x_1 + x_2}{n_1 + n_2} \), where \( x_1 = 13 \), \( x_2 = 16 \), \( n_1 = n_2 = 200 \).\[ p = \frac{13 + 16}{200 + 200} = \frac{29}{400} = 0.0725 \]The test statistic is given by:\[ Z = \frac{\hat{p}_1 - \hat{p}_2}{\sqrt{p(1-p)\left(\frac{1}{n_1} + \frac{1}{n_2}\right)}} \]Using \( \hat{p}_1 = \frac{13}{200} = 0.065 \) and \( \hat{p}_2 = \frac{16}{200} = 0.08 \), we compute:\[ Z = \frac{0.065 - 0.08}{\sqrt{0.0725(1-0.0725)\left(\frac{1}{200} + \frac{1}{200}\right)}} \]\[ Z \approx \frac{-0.015}{\sqrt{0.0725 \times 0.9275 \times 0.01}} \approx \frac{-0.015}{0.0262} \approx -0.572 \]

Make the Decision

We compare the calculated test value \( Z = -0.572 \) with the critical values \( \pm 1.96 \). Since \( -0.572 \) does not fall within the rejection region (i.e., it is not less than -1.96 or greater than 1.96), we fail to reject the null hypothesis.

Summarize the Results

Based on our calculations and the comparison between the test statistic and critical values, there is not enough statistical evidence to conclude that there is a significant difference in the proportion of students bullied in private compared to public schools at the 0.05 significance level.

Key concepts

Critical Values

Critical values are the threshold values that divide the regions in which the null hypothesis will be rejected from the regions in which it will not be rejected. In hypothesis testing, these values help determine whether the results of an experiment or survey are statistically significant. In other words, they help us decide if what we observed in a study is unlikely to have occurred by chance alone.
For a two-tailed test, the critical values can be found using the standard normal distribution, commonly referred to as the Z-distribution. At a 0.05 significance level, which is common for such tests, the critical values are typically found at the Z-scores of

• -1.96
• +1.96

This means that for the test results to be considered statistically significant, the calculated Z-value needs to fall either below -1.96 or above 1.96.

Z-distribution

The Z-distribution, also known as the standard normal distribution, is a probability distribution that is symmetrically centered around zero. It has a mean of 0 and a standard deviation of 1, which makes it a very useful tool in statistics. For hypothesis testing of differences between proportions, like in the study of bullying between private and public schools, the Z-distribution is utilized to find the probability that a result as extreme or more extreme than the observed one would occur under the null hypothesis.
When comparing proportions using a Z-test:

• The null hypothesis generally states that there is no difference between the proportions.
• If the computed Z-value is greater than the critical value, the null hypothesis is rejected, indicating a statistically significant difference.
• If not, we fail to reject the null hypothesis.

Proportions

Proportions are used in statistics to represent fractions, percentages, or parts of a whole. They are particularly handy when comparing two different groups, as they allow for a standard of comparison. In the example of bullying, the proportion of bullied students is the number of students who reported being bullied divided by the total number of students in the sample. For instance:

• In private schools, the proportion of bullied students is given by: \[ \hat{p}_1 = \frac{13}{200} = 0.065 \]
• In public schools, it is: \[ \hat{p}_2 = \frac{16}{200} = 0.08 \]

These sample proportions are crucial for calculating the test statistic, which helps determine if the observed difference between the groups could be due to chance.

Null Hypothesis

The null hypothesis (\( H_0 \)) is a statement that indicates no effect or no difference in the context of statistical testing. In many experimental and observational studies, the goal is to either reject or fail to reject this statement. In the bullying example:

• The null hypothesis claims there is no difference in the proportions of bullied students between private and public schools. Mathematically, this is written as: \[ H_0: p_1 = p_2 \]
• The alternative hypothesis (\( H_a \)) suggests a difference in the proportions, i.e., \[ H_a: p_1 eq p_2 \]

To determine whether we reject the null hypothesis, we compare the calculated test statistic against the critical values defined previously. If it falls outside the range defined by these critical values, it suggests a statistically significant result, leading to the rejection of the null hypothesis.