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In \(2015,\) a survey of firstyear university students in Brazil was conducted to determine if they knew how to activate the Mobile Emergency Attendance Service (MEAS). Of the 1038 respondents (59.5\% studying biological sciences, \(11.6 \%\) physical sciences, and \(28.6 \%\) humanities) \(, 54.3 \%\) students of nonbiological subjects \((n=564)\) knew how to activate the MEAS as compared to \(61.4 \%\) students of biological sciences \((n=637)\). (Source: https://www.ncbi.nlm.nih.gov/ \(\mathrm{pmc/articles} / \mathrm{PMC} 4661033 /)\) a. Estimate the difference between the proportions of students of biological sciences and nonbiological subjects who know how to activate the MEAS and interpret. b. Find the standard error for this difference. Interpret it. c. Define the two relevant population parameters for comparison in the context of this exercise. d. Construct and interpret a \(95 \%\) confidence interval for the difference in proportions, explaining how your interpretation reflects whether the interval contains \(0 .\) e. State and check the assumptions for the confidence interval in part d to be valid.

Step by step solution

01

Define Proportions

First, define the sample proportions for students knowing how to activate the MEAS. For biological sciences, the proportion \( p_1 \) is \( 0.614 \). For nonbiological subjects, the proportion \( p_2 \) is \( 0.543 \).

02

Estimate the Difference in Proportions

Calculate the difference between the two proportions: \( p_1 - p_2 = 0.614 - 0.543 = 0.071 \). This difference represents a 7.1% higher proportion of biological sciences students knowing how to activate MEAS compared to nonbiological students.

03

Calculate the Standard Error

The formula for the standard error (SE) of the difference between two proportions is \[ SE = \sqrt{\frac{p_1(1-p_1)}{n_1} + \frac{p_2(1-p_2)}{n_2}} \] where \( n_1 = 637 \) and \( n_2 = 564 \). Plug in the values: \[ SE = \sqrt{\frac{0.614 \times 0.386}{637} + \frac{0.543 \times 0.457}{564}} \approx 0.0275. \] This standard error quantifies the variability of our difference estimate.

04

Define Relevant Population Parameters

In this context, the two relevant population parameters are the proportion of all biological sciences students who know how to activate the MEAS and the proportion of all nonbiological subjects students who do.

05

Construct a 95% Confidence Interval

Use the formula for a confidence interval for the difference in proportions: \[ (p_1 - p_2) \pm Z \times SE \] where \( Z = 1.96 \) for a 95% confidence level. The interval is \[ 0.071 \pm 1.96 \times 0.0275 \approx (0.016, 0.126). \] This interval suggests that we are 95% confident that the true difference in proportions is between 1.6% and 12.6%.

06

Check Assumptions for the Confidence Interval

The assumptions for the validity of a confidence interval for difference in proportions include: 1) Random sampling, and 2) Adequate sample sizes so that each group has at least 10 successes and 10 failures. Both conditions are satisfied in this survey as per the data provided.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Confidence Interval

Confidence intervals provide a range of values that are believed to capture an unknown population parameter. In practice, they offer an interval of plausible values based on sample data, essentially providing an estimated range where the true value likely falls.

When we construct a confidence interval for the difference in proportions, it informs us how much one group's characteristic differs from another. In this particular exercise, we are comparing students from biological sciences with those from nonbiological subjects regarding their knowledge of activating the MEAS.

The formula for a confidence interval includes the sample estimate (the observed difference) and reflects the standard error of the estimate, expanded or contracted according to the desired confidence level:

• The midpoint of the interval is the estimated proportion difference.
• The margin of error adjusts the interval width and is calculated as the product of the standard error and the critical value from the standard normal distribution (e.g., 1.96 for 95% confidence).
• Intervals that do not include zero indicate a statistically significant difference.

In this case, the 95% confidence interval from 1.6% to 12.6% suggests that we can be 95% sure the true difference in proportions is in this range, implying biological sciences students are indeed more knowledgeable about MEAS activation.

Proportions

A proportion indicates the fraction of the whole that meets a specific criterion, usually expressed as a percentage. In statistical studies, proportions allow us to compare different categories or groups in a meaningful way.

In the given scenario, we are looking at two proportions:

• The proportion of biological sciences students who know how to activate the MEAS: \( p_1 = 0.614 \).
• The proportion of nonbiological subjects students: \( p_2 = 0.543 \).

These figures are at the heart of the analysis since the task involves comparing their difference to infer the students’ knowledge disparity.

Understanding proportions is essential since:

• They provide a simple way to quantify characteristics in a sample or population.
• Differences in proportions can offer insights that lead to actionable conclusions or interventions.
• They serve as the foundation for calculating the difference used in confidence intervals and standard errors.

By effectively analyzing proportions, you can draw significant conclusions about the population being studied.

Standard Error

The standard error (SE) provides a measure of the variability of a statistic. It indicates how much the sample statistic, such as the difference in proportions, is expected to vary if different samples are taken from the population.

The formula for calculating the standard error of the difference between two proportions ensures we consider both sample sizes and variabilities in the two groups. The formula is:\[SE = \sqrt{\frac{p_1(1-p_1)}{n_1} + \frac{p_2(1-p_2)}{n_2}}\]Where:

• \( p_1 \) and \( p_2 \) are the sample proportions of the two groups.
• \( n_1 \) and \( n_2 \) are the sample sizes of those groups.

The exercise provides an SE of approximately 0.0275 for the difference. This value of SE implies that there is inherent variability of 2.75% in the estimated difference under repeated sampling.

Knowing SE helps in several ways:

• It aids in constructing confidence intervals by providing the margin of error.
• A smaller SE indicates more precise estimates of our statistic.
• Understands the reliability of our estimates, where less variability approximates closer to true population differences.

Overall, the SE plays a critical role in interpreting the spread and reliability of estimated results.

Population Parameters

Population parameters represent characteristics of an entire population, providing key insights into general trends and truths for that group.

In this exercise, the focus is on defining two relevant parameters:

• The first parameter is the true proportion of all biological sciences students who can activate the MEAS.
• The second is the corresponding proportion among all nonbiological subjects students.

These parameters are the ultimate target of our inference because they describe the population characteristics we aim to understand.

The role of these parameters in our analysis includes:

• Serving as benchmarks for practical decisions or interventions.
• Providing the basis for constructing inferential statistics like confidence intervals.
• Serving as a means to translate findings from smaller samples to broader contexts.

Thus, understanding population parameters is crucial since they allow us to transform sample data into meaningful insights applicable to larger groups.