91Ó°ÊÓ

According to indexmundi.com, the rates of literacy in five countries in the Middle East with the lowest rates of literacy are given in the table. $$ \begin{array}{ll} \text { Afghanistan } & 28 \\ \hline \text { Pakistan } & 55 \\ \hline \text { Yemen } & 65 \\ \hline \text { Iraq } & 79 \\ \hline \text { Syria } & 84 \\ \hline \end{array} $$ a. Find and report the mean rate of literacy per country in context: The mean percentage of literates in these five countries is (Report the percentage to the nearest tenth.) b. Sketch a dotplot of the data and mark the location of the mean. You can use a triangle under the axis \((\mathbf{A}\), like a fulcrum) to mark the location; it should be at the balance point. c. Find and report the standard deviation of the rate of literates per country in context; round to the nearest hundredth. d. Which of these rates is farthest from the mean and therefore contrib- utes most to the standard deviation?

Step by step solution

01

Compute Mean Literacy Rate

Sum up the literacy rates of the five countries, i.e., \(28+55+65+79+84\) and then divide by the total number of countries, which is 5.

02

Draw a Dotplot

Create a scale along a straight line, preferably scaled to match the literacy rates range (i.e., 20 to 84). Plot the given values on this line as discrete dots. Mark the previously calculated mean as a triangle on the axis.

03

Calculate Standard Deviation

The formula for standard deviation \(s\) is \(s=\sqrt{\frac{\sum_{i=1}^{n} (x_{i}-\bar{x})^2}{n-1}}\) where \(x_{i}\) are the literacy rates, \(\bar{x}\) is the calculated mean, and \(n\) is the number of countries. Substitute the given values into the formula and calculate the standard deviation.

04

Identify the Rate Farthest from the Mean

Compute the differences between the mean and each of the rates. The rate with the greatest difference from the mean contributes most to the standard deviation.

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Key Concepts

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

Mean Calculation

The mean is a measure of central tendency that provides us with an average value. In this context, we are finding the average literacy rate among the five Middle Eastern countries listed: Afghanistan, Pakistan, Yemen, Iraq, and Syria. To calculate the mean, we begin by summing all the literacy rates:

• Afghanistan: 28%
• Pakistan: 55%
• Yemen: 65%
• Iraq: 79%
• Syria: 84%

Adding these up gives us 311. The next step is to divide this sum by the number of countries, which is 5. The calculation is as follows:\[\text{Mean} = \frac{28 + 55 + 65 + 79 + 84}{5} = \frac{311}{5} = 62.2\%\]This result tells us that on average, around 62.2% of individuals in these countries can read and write.

Standard Deviation

Standard deviation is a statistical measure that indicates the amount of variation or dispersion in a set of values. A high standard deviation suggests that the literacy rates are spread out over a wide range,while a low standard deviation indicates that they are closely clustered around the mean.Let's calculate it using the formula:\[s=\sqrt{\frac{\sum_{i=1}^{n} (x_{i}-\bar{x})^2}{n-1}}\]Here, \(x_i\) represents each literacy rate, \(\bar{x}\) is the mean (62.2), and \(n\) is the total number of rates (5).First, subtract the mean from each rate and square the results:

• \((28 - 62.2)^2 = 1163.84\)
• \((55 - 62.2)^2 = 51.84\)
• \((65 - 62.2)^2 = 7.84\)
• \((79 - 62.2)^2 = 282.24\)
• \((84 - 62.2)^2 = 475.24\)

The sum of these squares is 1980. The standard deviation calculation becomes:\[s = \sqrt{\frac{1980}{4}} = \sqrt{495} \approx 22.25\%\]This indicates that most countries' literacy rates deviate from the mean by about 22.25%.

Dotplot Representation

A dotplot is a simple visual representation to display data, particularly useful for small datasets like this one. It allows us to see the distribution and frequency of data points in a clear, concise manner.To construct a dotplot for the literacy rates, you should follow these steps:

• Draw a horizontal line and label it with an appropriate scale covering the range from 20% to 84%.
• Place a dot above the corresponding value for each literacy rate on the line:
• 28%: Place one dot.
• 55%: Place one dot.
• 65%: Place one dot.
• 79%: Place one dot.
• 84%: Place one dot.

Mark the mean, 62.2%, with a different symbol—such as a triangle (\( \mathbf{A} \))—to showcase its position. This dotplot helps clearly visualize that while one country's literacy rate (Afghanistan) is much lower, the rest vary more closely around the mean.