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Chapter 10: Problem 7

Effects of Parental Education on Boys' Education (Example 1) A study done by the IZA Journal of Labour Economics in the United Kingdom in 2013 compared 15 -year-old boys 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 boys' education and parental education. \(\begin{array}{lcc} & \text { Educated Parents } & \text { Uneducated Parents } \\ \text { Studying } & 42 & 23 \\ \text { Not Studying } & 14 & 8\end{array}\) 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 boys who are studying. c. Find the expected number of boys having educated parents who would study, if the variables are independent. Multiply the proportion overall that were studying times the number of boys having educated parents. Do not round off to a whole number. Round to two decimal digits. d. Find the other expected counts using your knowledge so that the expected counts must add to the row and column totals. Report them in a table with the same orientation as the one given for the data.

Short Answer

Expert verified
a. The row, column, and grand totals are: Educated Parents - 42 , Uneducated Parents - 23, Studying - 65, Not Studying- 22, Grand Total - 87. b. The percentage of boys studying is approximately 74.71%. c. Assuming variable independence, the expected number of boys with educated parents who would study is approximately 33.33. d. The remaining expected counts are: Studying with Uneducated Parents - 31.67, Not Studying with Educated Parents - 14.00, Not Studying with Uneducated Parents - 7.88.

Step by step solution

01

Find the row, column, and grand totals

The row total is calculated by adding the values in each row. Similarly, the column total is calculated by adding each column. Grand total is the sum of all values in the table. In this case, the totals are:\n\nFor Educated Parents: \(42 (studying) + 14 (not studying) = 56\)\nFor Uneducated Parents: \(23 (studying) + 8 (not studying) = 31\)\nFor Studying Boys: \(42 (educated parents) + 23 (uneducated parents) = 65\)\nFor Not Studying Boys: \(14 (educated parents) + 8 (uneducated parents) = 22\)\nGrand Total = \(56 (educated parents) + 31 (uneducated parents) = 87\)
02

Calculate the percentage of boys who are studying

The proportion (percentage) of boys who are studying is calculated by taking the total number of studying boys and dividing it by the grand total:\n\nPercentage of Studying Boys = \((65 Studying Boys / 87 Grand Total)*100 = 74.71% \)approx
03

Calculate the expected number of boys having educated parents who would study

Assuming the variables are independent, the expected count for studying boys with educated parents is calculated by multiplying the proportion of studying boys by the total number of boys having educated parents. Proportional calculation is: Studying Boys / Grand Total:\n\nExpected Number = \((65 (Studying Boys)/ 87 (Grand Total)) * 56 (Boys with Educated Parents) = 33.33 approx\).
04

Find other expected counts

Using the independence assumption - expected counts for each category can be calculated as: Expected Count = (Row Total * Column Total) / Grand Total.\n\nExpected count for:\n- Studying boys with Uneducated Parents = \((65 Studying Boys * 31 Uneducated Parents) / 87 (Grand Total) = 31.67\)\n- Non-studying boys with Educated Parents = \((22 Non-Studying Boys * 56 Educated Parents) / 87 (Grand Total) = 14.00\)\n- Non-studying boys with Uneducated Parents = \((22 Non-Studying Boys * 31 Uneducated Parents) / 87 Grand Total = 7.88 \)

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Key Concepts

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

Parental Education Impact
The influence of parental education on a child's academic progress is a significant factor in educational research and policy making. The above-mentioned study highlights the correlation between the educational level of parents and the educational status of their children, specifically 15-year-old boys. High parental education often equates to a more enriched educational environment at home, better academic support, and greater stress on the importance of education, all of which can influence a child's decision to continue studying.

Understanding this relationship helps in developing targeted interventions that support children from less-educated family backgrounds. For instance, educational programs could be crafted to provide additional support to boys whose parents have lower levels of education, with the goal of reducing drop-out rates and enhancing academic performance. Such insights are invaluable for educators and policymakers who work towards equitable education systems where every child has the chance to succeed.
Expected Count Calculation
Expected count calculation in statistical data analysis is a method used to estimate the frequency of occurrences in a given category if there were no association between the variables. It is a crucial component in performing the chi-square test of independence. The step-by-step solution provided illustrates how to calculate the expected counts given the assumption of independence between parental education levels and boys' educational status.

It is important to note that the expected count is calculated using the formula:

\[\text{Expected Count} = \left(\frac{\text{Row Total} \times \text{Column Total}}{\text{Grand Total}}\right)\]

This formula helps identify whether the observed frequencies significantly deviate from the expected frequencies under the assumption of independence. If the differences are substantial, it suggests that there might be a relationship between the two variables being studied. Finally, rounding to two decimal places ensures precision without introducing significant rounding errors, which is key in any statistical analysis.
Independence in Statistics
Independence in statistics is a foundational concept that asserts two variables are unrelated to each other; that is, the occurrence of one does not affect the probability of the occurrence of the other. To establish independence formally, statistical tests such as the chi-square test are used, which compare observed counts to expected counts calculated under the assumption of independence.

When the observed data is markedly different from what we would expect if the variables were independent, we consider the variables to be dependent or associated with each other. This is essential in determining the presence of a statistically significant relationship, as seen in the study on parental education and boys' education. In our exercise, by calculating expected counts, we can compare them with the observed counts to assess the independence of boys' studying status from their parents' education level. Such statistical evaluations are critical in research that informs policy decisions and educational strategies.

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Most popular questions from this chapter

Literacy Rates World literacy rates for individuals of 15 years of age or older are given in the data table as a percentage. Give two reasons why a chi- square test is not appropriate for this set of data.

Odd-Even Formula A survey was taken of a random sample of people noting their gender and asking whether they agreed with the Odd-Even Formula (OEF) to control the alarming levels of air pollution. Minitab results are shown. \(=1\) 2 Chi-square Test for Association: Opinion, Gender ciat:1 Rows: Ooinion Columns: Gender $$ \begin{array}{rr}\text { Male Female } \\ \text { Disagree } & 42 & 44 \\\ 42.17 & 43.83 \\ \text { Agree } & 11.4 & 86 \\ & 110 & 14.17 & \\ & 109.83 & \\ \mathrm{All} & & \\ & 152 & 158 & 310 \\ \text { Cel1 Contents: } & \text { Count } \\ & \text { Expected count } \\ \text { Pearson Chi-Square }=0.002, \mathrm{DF}=1, \text { p-value }=0.966\end{array} $$ a. Find the percentage of men and women in the sample who agreed with the OEF method, and compare these percentages. b. Test the hypothesis that opinions about OEF and gender are independent using a significance level of \(0.05\). c. Does this suggest that men and women have significantly different views about the OEF method?

Antibiotic or Placebo A large number of surgery patients get infections after surgery, which can sometimes be quite serious. Researchers randomly assigned some surgery patients to receive a simple antibiotic ointment after surgery, others to receive a placebo, and others to receive just cleansing with soap. If we wanted to test the association between treatment and whether or not patients get an infection after surgery, would this be a test of homogeneity of of independence? Explain. (Source: Hospitals could stop infections by tackling bacteria patients bring in, studies find. New York Times, January \(6,2010 .)\)

The Perry Preschool Project discussed in Exercises \(10.23-10.25\) found that 8 of the 58 students who attended preschool had at least one felony arrest by age 40 and that 31 of the 65 students who did not attend preschool had at least one felony arrest (Schweinhart et al. 2005). a. Compare the percentages descriptively. What does this comparison suggest? b. Create a two-way table from the data and do a chi-square test on it, using a significance level of \(0.05 .\) Test the hypothesis that preschool attendance is associated with being arrested. c. Do a two-proportion \(z\) -test. Your alternative hypothesis should be that preschool attendance lowers the chances of arrest. d. What advantage does the two-proportion \(z\) -test have over the chisquare test?

Colored Vegetables and Stroke A study of colored vegetables and the risk of stroke was done. Although the investigators did not see any effect on stroke of consumption of green, orange, red, yellow, or purple vegetables, they concluded that "High intake of white fruits and vegetables may protect against stroke." Suppose that for each pair of color groups, we wished to test whether the stroke risk was different. For instance, is it different for green vs. orange? For green vs. red? Suppose there are 10 different pairs to be compared. Suppose that, for each pair, we perform a hypothesis test with a significance level of \(10 \%\). Assume that in truth, there are no differences between any of the pairs. By chance alone, how many of the hypothesis tests would you expect to appear significant (and thus lead us to mistakenly believe that there was a difference)? (Source: Oude Griep, et al. 2011. Colors of fruit and vegetables and 10-year incidence of stroke. Stroke \(42(11), 3190-3195 .\) )
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