Question

Public Agenda conducted a survey of 1,379 parents and 1,342 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 these were essential. The two samples - parents and students- were independently selected random samples. Using a confidence level of \(95 \%\) estimate the difference between the proportion of students who think that understanding science and having math skills are essential and this proportion for parents.

Short answer

The 95% confidence interval for the difference of proportions between students and parents who believe that understanding science and having strong math skills is essential.

Step-by-step solution

Find the Proportions

First of all, compute the proportions for both students and parents. Given, the survey was conducted among 1,379 parents and 1,342 students. It is said that \(50 \% \) of students and \( 62 \% \) of parents believe that understanding science and having math skills are essential. Then, Proportion of students \(p_1 = 0.50 \) and proportion of parents \(p_2 = 0.62 \). Also, the number of students \(n_1 = 1342 \) and the number of parents \(n_2 = 1379 \).

Compute Difference of Proportions

Subtract \(p_1\) from \(p_2\) to find the difference between the two proportions. \(p = p_2 - p_1 = 0.62-0.50 = 0.12\).

Find the Standard Error

Compute the standard error (SE) which is the square root of \( p_1(1-p_1)/n_1 + p_2(1-p_2) /n_2 \), giving \( SE = \sqrt{0.5*(1-0.5)/1342 + 0.62*(1-0.62)/1379} \).

Calculate the Confidence Interval

Given a 95% confidence level, the Z value is 1.96 (from the standard normal distribution table). The 95% Confidence Interval for the difference of proportions is \( p \pm Z * SE \), where Z is the Z-value. Calculating this gives the Confidence Interval.

Key concepts

Confidence Interval

Understanding a confidence interval is critical when analyzing the results of a survey like the one conducted on students and parents regarding the importance of science and math skills. A confidence interval offers a range of possible values for the parameter of interest — in this case, the difference in the proportions of students and parents who believe these skills are essential. In the provided exercise, a 95% confidence interval was used. This means we can be 95% confident that the true difference in proportions lies within this calculated interval. To find this interval, we take the difference between the two proportions and adjust it by the amount of uncertainty, which is determined by the standard error and a z-value reflecting our confidence level.

Standard Error

The concept of standard error (SE) is essential in survey data analysis as it helps measure the variability or precision of the sample statistic. It's why the term 'error' should not be misunderstood as a mistake but rather as an indicator of uncertainty or variation. In the context of our exercise, the standard error quantifies the variability in the estimated difference between the two proportions being compared. Importantly, a lower standard error suggests that our estimate of the difference is more precise.

The SE is computed using the formula:
\[ SE = \sqrt{\frac{p_1(1-p_1)}{n_1} + \frac{p_2(1-p_2)}{n_2}} \]
where \(p_1\) and \(p_2\) are the sample proportions, and \(n_1\) and \(n_2\) are the sample sizes. Notably, the larger the sample size, the smaller the standard error, indicating a more reliable estimate.

Survey Data Analysis

Analyzing survey data often requires different statistical tools to interpret the collected data effectively. In the exercise, the analysis involves comparing two proportions from independent samples — students and parents. When processing survey data, we must ensure the samples are representative and that random sampling is applied, as this affects the validity of our conclusions. After validating our data collection, we use statistical methods, such as the calculation of proportions, standard errors, and confidence intervals, to infer about the entire population from our sample.

Proportion Calculation

The proportion calculation is quite foundational in statistics, as seen in our exercise. It refers to the ratio of the number of times an outcome occurs to the total number of trials or sample size. In the context of the survey, the proportion helps us determine what fraction of the respondents — students and parents — views science and math skills as essential. To calculate each proportion, divide the number of individuals with the concerned opinion by the total number surveyed, giving us \(p_1\) and \(p_2\). These proportions serve as estimates of the population parameters and form the basis for further statistical analysis, like evaluating the difference between the two groups.

Z-Value

The z-value plays a pivotal role when constructing confidence intervals within survey data analysis. It is a statistic that represents how many standard deviations an element is from the mean. Specifically, for a given confidence level—95% in the exercise—the z-value maps to that probability on the standard normal distribution. Often known as a 'z-score', this number is used to multiply the standard error in the confidence interval formula, signifying the number of standard errors away from the observed difference you must go to achieve the desired confidence in your interval estimate. For a 95% confidence interval, the z-value is commonly 1.96.

Statistics Education

Imparting the fundamentals of statistics education is invaluable, especially when tackling real-world problems such as analyzing survey data. Creating an educational approach that demystifies complex concepts and formulas into straightforward, digestible pieces helps students understand and apply statistical methods with confidence. In the statistics education process, exercises like the one given provide a practical application, reinforcing concepts such as proportion calculations, standard error, and the interpretation of confidence intervals, all of which are everyday tools in a statistician's arsenal.