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Chapter 10: Problem 8

Effects of Parental Education on Girls' Education Refer to Exercise \(10.7 .\) This data table compares 15 -year-old girls who are either attending school or have dropped out, in order to understand the impact of parental education on them. The table shows the relationship between girls' education and parental education. \begin{tabular}{lcc} & Educated Parents & Uneducated Parents \\ \hline Studying & 13 & 68 \\ Not Studying & 11 & 6 \end{tabular} a. Find the row, column, and grand totals, and prepare a table showing these values as well as the counts given. b. Find the percentage of girls who are studying. c. Find the expected number of girls having educated parents who would study, if the variables are independent. Multiply the proportion overall that were studying times the number of girls having educated parents. Do not round off to a whole number. Round to two decimal digits. d. Find the other expected counts. Report them in a table with the same orientation as the one for the data.

Short Answer

Expert verified
a. Updated Table: \[\begin{tabular}{lcc} & Educated Parents & Uneducated Parents & Totals \\ \hline Studying & 13 & 68 & 81 \\ Not Studying & 11 & 6 & 17 \\ \hline & 24 & 74 & 98 \end{tabular}\] b. Percentage of girls studying: 82.65%. c. Expected count (educated, studying): 19.82 d. Other expected counts: studying with uneducated parents (61.11), not studying with educated (4.18) and uneducated parents (12.89).

Step by step solution

01

Compute Row, Column, and Grand Totals

For this, add the values horizontally for row totals, and vertically for column totals. Finally, add up these totals to get the grand total. So, the number of girls with educated parents is \(13 + 11 = 24\) and with uneducated parents is \(68 + 6 = 74\). Hence, the total number of girls studying is \(13 + 68 = 81\) and not studying is \(11 + 6 = 17\). Grand total is \(24 + 74 = 98\).
02

Calculate the Percentage of Studying Girls

The proportion of girls studying can be computed as the total number of girls studying divided by the grand total times 100. That is \((81/98) * 100 = 82.65\%\).
03

Find Expected Counts Under Independence

First, find the proportion of girls that are studying, which is \(81/98 = 0.826\). Then, this is multiplied by the total number of girls with educated parents (24), to get an expected count of \(0.826 * 24 = 19.82\).
04

Compute Other Expected Counts

Now, compute the expected counts for the remaining categories by multiplying the total number of girls in each category by the studying and non-studying proportions. This gives \(0.826 * 74 = 61.11\) (studying, uneducated parents), \(0.174 * 24 = 4.18\) (not studying, educated parents), and \(0.174 * 74 = 12.89\) (not studying, uneducated parents).

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Key Concepts

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

Statistical Data Analysis
Statistical data analysis involves interpreting a collection of data to understand patterns, relationships, and insights within. For instance, in examining the impact of parental education on girls' education, raw data is collected reflecting the number of girls studying or not based on their parents' education. Analyzing this data provides a deeper understanding of the influence that an educated home environment has on a girl's educational pursuits.

To begin, we sum up the number of girls based on their education status and parental education level, identifying totals for each category and overall. This comprehensive approach allows us to view the macro-level effect parental education might have on the daughters鈥攚hether it encourages school attendance or correlates with higher dropout rates.

Understanding Raw Data

Initiating the analysis, we organize the raw data, which would show, for example, that out of 98 girls, 13 with educated parents are studying, and 11 are not. Meanwhile, 68 girls with uneducated parents are in school, whereas 6 are not. These figures lay the foundation for deeper statistical analysis.
Expected Counts Under Independence
The concept of expected counts under independence in statistics is fundamental when we want to determine whether two variables are related. Independence implies that the occurrence of one event does not influence the probability of the occurrence of the other event.

In our exercise, we seek to figure out what the numbers would look like if parental education had no effect on whether girls were studying or not. To calculate this, we use overall proportions to estimate expected counts. Essentially, one multiplies the total proportion of girls studying regardless of parental education by the number of girls in each parental education category.

Determining Independence

Here, if 82.65% of all girls are studying, we would expect 82.65% of the girls from both educated and uneducated parents to be studying if parental education and studying were independent. These calculations offer a theoretical distribution that can then be compared to actual data to identify any significant differences, potentially indicating dependency. Using this method, we can explore the relationship between parental education levels and their children's education in a quantifiable manner.
Calculating Percentages in Statistics
Calculating percentages is a staple in statistical analysis. It helps us understand how a part relates to the whole in terms of proportion. When examining the percentage of girls who are studying, we relate the number of girls attending school to the total population considered.

The formula for calculating a percentage is straightforward: you divide the count of the subset by the total count and multiply the result by 100 to convert it to a percentage. In our exercise, dividing the number of studying girls (81) by the total number of girls (98) and then multiplying by 100 yields approximately 82.65%.

Interpreting Percentages

This percentage is more than just a number; it gives context. In our context, it signifies that a sizeable majority of the girls in the data set are attending school. Such calculations enable us to express data in a universally understood format, facilitating comparisons and further analysis.

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