Question

Public Expenditure on Education Figure 2.27 shows the public expenditure on education as percentage of Gross Domestic Product (GDP) for all countries. \(^{42}\) The mean expenditure is \(\mu=4.7 \%\) and the standard deviation of the expenditures is \(\sigma=2 \% .\) The data are stored in EducationLiteracy. (a) The United States spends \(5.2 \%\) of it's GDP on education. Without doing any calculations yet, will the \(z\) -score for the US be positive, negative, or zero? Why? (b) Calculate the \(z\) -score for the US. (c) There are two high outliers; Lesotho (a small country completely surrounded by South Africa) spends \(13 \%\) of it's GDP on education and Cuba spends \(12.8 \%\). Equatorial Guinea spends the lowest percentage on education at only \(0.8 \% .\) Calculate the range. (d) The five number summary for this data set is \((0.8,3.2,4.6,5.6,13) .\) Calculate the IQR.

Short answer

The predicted z-score for the US is positive. The calculated z-score for the US is 0.25. The range for the given data set is 12.2%. The interquartile range (IQR) is 2.4%.

Step-by-step solution

Predict the Nature of the Z-score

Since the mean expenditure is 4.7% and the US spends 5.2% of its GDP on education, it can be predicted that the z-score for the US would be positive because the US expenditure is above the mean.

Calculate the Z-score for the US

Z-score is given by the formula \(Z = \frac{X - \mu}{\sigma}\) where: X = value we are comparing = 5.2µ = mean of the population = 4.7 σ = standard deviation of the population = 2 Plugging these values into the formula gives\(Z = \frac{5.2 - 4.7}{2} = 0.25\).

Calculate the Range

The range of a data set is the difference between the highest and the lowest value in the set. Given that the highest value is 13% (Lesotho) and the lowest is 0.8% (Equatorial Guinea), the range can be calculated asRange = maximum value - minimum value = 13 - 0.8 = 12.2.

Calculate the Interquartile Range (IQR)

The IQR is the range of the middle 50% of the data. It's the difference between the third quartile (Q3) and the first quartile (Q1). From the five number summary given, Q3 = 5.6 and Q1 = 3.2. Therefore, IQR = Q3 - Q1 = 5.6 - 3.2 = 2.4.

Key concepts

Gross Domestic Product and education

When discussing the economic development and priorities of a nation, the link between Gross Domestic Product (GDP) and education is pivotal. GDP, which is the total value of all the goods and services produced over a specified time period, often serves as an indicator of a country's economic health. Investing a portion of GDP in education reflects a nation's commitment to developing human capital. The benefits of such public expenditure on education are manifold: it can drive economic growth, foster innovation, and equip individuals with the skills necessary to thrive in the workforce.Higher education spending, as a percentage of GDP, could imply a country's forward-looking stance on knowledge-based economies. However, the optimal percentage is a subject of debate among economists and educators. Higher spending doesn't always equate to better educational outcomes; effectiveness and efficiency of spending also play crucial roles.

z-score calculation

Understanding Z-scores

A z-score represents the number of standard deviations a data point is from the mean. It is a tool that enables comparison of values from different sets of data or against a population mean.Calculating the z-score involves subtracting the mean from the data point in question and then dividing the result by the standard deviation. This is basically standardizing data.The formula for z-score is: \[ Z = \frac{X - \mu}{\sigma} \]

Interpreting Z-scores

A positive z-score indicates that the data point is above the mean, while a negative z-score shows it's below the mean. A z-score of zero implies that the data point is exactly at the mean. Z-scores also help identify outliers; points with z-scores that are too high or too low compared to the rest may be considered unusual.

statistical range

In statistics, the range of a dataset is the simplest measure of variability. It is calculated by finding the difference between the largest and smallest values. The formula is: \[ \text{Range} = \text{Maximum value} - \text{Minimum value} \]While the range gives a quick sense of the spread, it has its limitations as it only takes two points in the dataset into consideration, ignoring the rest of the data composition. This can sometimes be misleading, especially in datasets with outliers that can skew the range considerably. Despite these limitations, the range is still commonly used because of its simplicity.

Interquartile Range (IQR)

What is the IQR?

The Interquartile Range (IQR) is a measure of statistical dispersion and is considered a more robust and informative metric than the range. It describes the middle 50% of data, providing a sense of the spread of the central portion of a dataset.To calculate the IQR, first identify the first quartile (Q1), which is the middle value between the smallest number and the median of the dataset, and the third quartile (Q3), which is the middle value between the median and the highest number of the dataset. Then, subtract Q1 from Q3:\[ IQR = Q3 - Q1 \]

Significance of the IQR

Unlike the range, the IQR is not as sensitive to outliers. It is often used alongside the median to build a boxplot, which graphically depicts the distribution of a dataset. The IQR can help detect outliers and is key in creating fencing criteria to classify extreme values in data.