Question

Public Agenda conducted a survey of 1379 parents and 1342 students in grades \(6-12\) regarding the importance of science and mathematics in the school curriculum (Associated Press, February 15,2006 ). It was reported that \(50 \%\) of students thought that understanding science and having strong math skills are essential for them to succeed in life after school, whereas \(62 \%\) of the parents thought it was crucial for today's students to learn science and higher-level math. The two samples-parents and students-were selected independently of one another. Is there sufficient evidence to conclude that the proportion of parents who regard science and mathematics as crucial is different than the corresponding proportion for students in grades \(6-12 ?\) Test the relevant hypotheses using a significance level of \(.05\).

Short answer

The decision to reject or not reject the null hypothesis depends on the test statistic and the chosen significance level. The calculation steps above will give the necessary values to make this decision.

Step-by-step solution

Define the Null and Alternative Hypotheses

The null hypothesis (H0) will be that there's no difference between the proportions of parents and students who see science and math as key for future success, i.e., the two proportions are equal. On the other hand, the alternative hypothesis (Ha) is that there is a difference, i.e., the proportions are not equal. Mathematically, they can be written as: H0: \(p1 = p2\) and Ha: \(p1 ≠ p2\). Where \(p1\) denotes the proportion of parents, and \(p2\) denotes the proportion of students.

Calculate the Pooled Sample Proportion

Then calculate the pooled sample proportion (p) considering both samples together. The pooled sample proportion is given by the total number of 'successes' (Parents and students who believe science and math are important) divided by the total sample size. \(p = (x1 + x2) / (n1 + n2)\), where x denotes the number of 'successes' and n is the sample size.

Calculate the Standard Error

The standard error for the proportion difference is calculated using the pooled sample proportion. The formula is: \( SE = \sqrt{p * ( 1 - p ) * [ (1/n1) + (1/n2)] }\).

Calculate the Test Statistic

Calculate the test statistic (z) needed to conduct the hypothesis test. It's the difference in sample proportions minus the difference in population proportions (which is zero under the null hypothesis), divided by the standard error: \(z = (p1 - p2) / SE\).

Making the Decision

Based on the calculated test statistic and the critical value from z-table for a 0.05 significance level (1.96 for a two-tailed test), we would reject the null hypothesis if our test statistic is less than -1.96 or greater than 1.96. If it falls between -1.96 and 1.96, we would not reject the null hypothesis.

Key concepts

Null and Alternative Hypothesis

Understanding the null and alternative hypotheses is crucial when performing hypothesis testing in statistics. The null hypothesis, denoted as H0, is a statement of no effect or no difference. It serves as a starting point for statistical testing and provides a benchmark for evaluating the likelihood of sample data if there was no effect. In the given exercise, the null hypothesis is that the proportion of parents and students who view science and math as crucial for future success is the same, or mathematically, H0: \( p1 = p2 \).

The alternative hypothesis, denoted as Ha, contradicts the null hypothesis and is what you aim to support with sample data. It proposes that there is an effect or a difference. For this scenario, the alternative hypothesis suggests there is a difference in the proportions, expressed as Ha: \( p1 eq p2 \). It's important for students to clearly define these hypotheses because they shape the structure of the hypothesis testing process and guide the interpretation of results.

Pooled Sample Proportion

In the context of comparing two proportions, the pooled sample proportion is a way to estimate the common proportion of success when the null hypothesis assumes the two population proportions are equal. This estimate is given by combining the successes from both samples and dividing by the total number of observations.

To compute the pooled sample proportion, we use the formula \( p = \frac{x1 + x2}{n1 + n2} \), where \( x1 \) and \( x2 \) represent the number of successes in each sample, and \( n1 \) and \( n2 \) are the sizes of the two samples respectively. This value serves as a basis for further calculations in the hypothesis test and provides a more stable estimate of the variance when the sample sizes are different.

Standard Error Calculation

The standard error (SE) measures the variability in the sampling distribution of a statistic; in this case, the difference between two sample proportions. The calculation of SE incorporates the pooled sample proportion and the sizes of both samples. The formula for the standard error of the difference in proportions is \( SE = \sqrt{p * ( 1 - p ) * [ (1/n1) + (1/n2)] } \).

This step is fundamental as it reflects how much sample proportions are expected to vary from the actual population proportion difference. A smaller SE indicates that the sample proportion is likely to be closer to the actual population proportion, leading to more reliable hypothesis testing.

Test Statistic

Once the standard error is calculated, the next step in hypothesis testing is to determine the test statistic, which helps in deciding whether to reject the null hypothesis. The test statistic compares the observed difference to the expected difference under the null hypothesis, standardized by the standard error. For comparing two proportions, the test statistic (z) is calculated using the formula: \( z = \frac{p1 - p2}{SE} \).

The value of the test statistic is a crucial component of hypothesis testing as it determines the p-value or allows for comparison with critical values from a statistical distribution (such as a z-table). The test statistic represents how many standard errors the observed difference is away from the hypothesized difference.

Significance Level Analysis

The significance level, commonly denoted as alpha (\( \alpha \)), is a threshold that statisticians set to determine whether the evidence is strong enough to reject the null hypothesis. In the problem at hand, a significance level of 0.05 indicates that there is a 5% chance of rejecting the null hypothesis when it is actually true (Type I error).

Comparing the calculated test statistic against the critical values associated with the set significance level assists in making the statistical decision. If the test statistic falls outside the range of the critical values (which is -1.96 to 1.96 for a two-tailed test at the 0.05 significance level), it suggests that the null hypothesis may not correctly represent the population, leading to its rejection. Significance level analysis informs us about the robustness and reliability of our statistical conclusions.