Question

Birth Order and College Success In a study of the relationship between birth order and college success, an investigator found that 126 in a sample of 180 college graduates were firstborn or only children. In a sample of 100 non graduates of comparable age and socioeconomic background, the number of firstborn or only children was \(54 .\) Estimate the difference between the proportions of firstborn or only children in the two populations from which these samples were drawn. Use a \(90 \%\) confidence interval and interpret your results.

Short answer

Question: Based on the given information, estimate the difference between the proportions of firstborn or only children in the college graduates and non-graduates populations, and interpret the results using a 90% confidence interval. Data provided: - Firstborn or only children in the college graduates sample: 126 - Total college graduates in the sample: 180 - Firstborn or only children in the non-graduates sample: 54 - Total non-graduates in the sample: 100

Step-by-step solution

Identify the data

First, let's identify the key data given in the problem: - Number of firstborn or only children in the college graduates sample: 126 - Total number of college graduates in the sample: 180 - Number of firstborn or only children in the non-graduates sample: 54 - Total number of non-graduates in the sample: 100

Calculate the proportion of firstborn/only children in each sample

Now, let's find the proportion of firstborn or only children in each sample by dividing the number of firstborn or only children by the total number of individuals in each respective sample: Proportion of firstborn/only children in the college graduates sample: \(P_1 = \frac{126}{180}\) Proportion of firstborn/only children in the non-graduates sample: \(P_2 = \frac{54}{100}\)

Calculate the difference between the sample proportions

Next, let's find the difference between the proportions of firstborn/only children in the college graduates and non-graduates samples: Difference of proportions: \(P = P_1 - P_2\)

Calculate the standard error

Now, we need to calculate the standard error (SE) of the difference in proportions using the following formula: \(SE = \sqrt{\frac{P_1(1-P_1)}{n_1} + \frac{P_2(1-P_2)}{n_2}}\) Where \(n_1\) is the number of college graduates (180), and \(n_2\) is the number of non-graduates (100).

Determine the critical value for a 90% confidence interval

Since we want to calculate a 90% confidence interval, we need to find the critical value (z-value) for this level of confidence. The z-value can be found in z-tables online or using statistical software. In this case, the z-value for a 90% confidence interval is approximately 1.645.

Calculate the margin of error

Now, we can find the margin of error (ME) for our confidence interval using the formula: \(ME = z * SE\) Where \(z\) is the z-value we calculated in the previous step (1.645).

Calculate the 90% confidence interval

Finally, we can calculate the 90% confidence interval for the difference between the proportions of firstborn/only children in the two populations by adding and subtracting the margin of error from the difference in sample proportions: Confidence interval: \((P - ME, P + ME)\)

Interpret the results

With the 90% confidence interval calculated, we can interpret the results. If the confidence interval includes 0, then there is no significant difference between the proportions of firstborn/only children in the two populations. However, if the interval does not include 0, we can conclude that there is a significant difference in the proportions of firstborn/only children between the college graduates and non-graduates populations at a 90% confidence level.

Key concepts

Birth Order and College Success

The notion that an individual's birth order may impact their educational attainment has intrigued social scientists and psychologists for decades. The idea is that firstborns and only children, due to receiving more attention and resources from their parents, might have an advantage when it comes to academic achievements such as college success.

In studies that explore this, researchers often compare the proportion of firstborn or only children among college graduates with those who did not complete college. To discern a verifiable pattern, the investigator must ensure that the samples are drawn from populations with comparable ages and socioeconomic backgrounds. The goal is to mitigate extraneous variables that could skew the results. The key lies in meticulously collecting data, thoughtfully analyzing it, and drawing conclusions that are not only statistically sound but also meaningful in the real-world context of family dynamics and educational outcomes.

Research on birth order and educational achievement, such as the one described in the exercise, provides valuable insights into how familial factors may influence one's educational trajectory. Understanding this dynamic can help in crafting educational policies and support systems tailored to the needs of different family structures.

Proportion of Firstborn or Only Children

Proportions are a fundamental concept in statistics used to represent a part of a whole. In the context of this exercise, the proportion refers to the fraction of firstborn or only children within the two comparison groups: college graduates and non-graduates. To calculate the proportion, one simply divides the number of firstborn or only children by the total number of individuals in each sample:

• For college graduates: \(P_1 = \frac{126}{180}\)
• For non-graduates: \(P_2 = \frac{54}{100}\)

These proportions are critical in assessing whether there is a notable difference between the two groups in terms of birth order. Only by comparing these proportions can we begin to understand if and how birth order might be linked to academic success. The exercise guides students in calculating these proportions and encourages them to contemplate their implications. An accurate calculation of proportions is essential for any further statistical analysis, including estimating confidence intervals.

Statistical Significance

Statistical significance is a crucial concept in hypothesis testing, indicating whether the observed difference between groups could be due to chance or if it reflects a true pattern in the population. In simple terms, it helps researchers determine if their findings are meaningful or not.

In the step-by-step solution, after calculating the margin of error and the confidence interval, we check to see if zero is included in that range. If zero is not within the confidence interval, the difference we observed is statistically significant at the chosen confidence level—in this case, 90%. In our example, this would suggest that the proportion of firstborn or only children between the college graduates and non-graduates samples is significantly different.

The interpretation of these results has far-reaching implications not just in academia but also in forming educational policies, supporting children's development, and understanding the socio-cultural dynamics that influence educational outcomes. It is important for students to grasp the weight of statistical significance to ensure that the conclusions drawn from their analyses are robust and reliable.