91Ó°ÊÓ

\(\mathbf{}\) o \(\mathbf{\nabla}\) The following data values are 1989 per capita expenditures on public libraries for each of the 50 states: \(\begin{array}{rrrrrrr}29.48 & 24.45 & 23.64 & 23.34 & 22.10 & 21.16 & 19.83 \\\ 18.01 & 17.95 & 17.23 & 16.53 & 16.29 & 15.89 & 15.85 \\ 13.64 & 13.37 & 13.16 & 13.09 & 12.66 & 12.37 & 11.93 \\ 10.99 & 10.55 & 10.24 & 10.06 & 9.84 & 9.65 & 8.94 \\ 7.70 & 7.56 & 7.46 & 7.04 & 6.58 & 5.98 & 19.81 \\ 19.25 & 19.18 & 18.62 & 14.74 & 14.53 & 14.46 & 13.83 \\ 11.85 & 11.71 & 11.53 & 11.34 & 8.72 & 8.22 & 8.13 \\ 8.01 & & & & & & \end{array}\) a. Summarize this data set with a frequency distribution. Construct the corresponding histogram. b. Use the histogram in Part (a) to find approximate values of the following percentiles: i. 50 th iv. 90 th ii. 70 th v. 40 th iii. 10 th

Step by step solution

01

Create Frequency Distribution

Sort the data in ascending or descending order. After arranging the data, find the range (maximum - minimum) which will give an idea about the size of the class interval for the frequency distribution. Then, divide the data set into class intervals, count the number of observations falling into each class, which will give the frequency distribution.

02

Construct a Histogram

With the frequency distribution ready, this step involves visualizing the data. The class intervals will be on the x-axis and the frequencies on the y-axis. Each class interval will be represented by a bar; the height of the bar corresponds to the frequency of that class.

03

Calculate Percentiles

Once we have the histogram, we can find the percentiles. For instance, the 50th percentile (also known as the median) represents the value below which 50% of the observations may be found. Similarly, for 70th, 90th, 40th, and 10th percentiles. Use the formula for percentile calculation: \(P_k = \frac{k(N+1)}{100}\) where \(P_k\) is the kth percentile and N is the total number of observations.

04

Interpret Percentiles

Interpreting the percentiles gives us the approximate values of the given percentiles in terms of expenditure. This informs us about the data distribution, revealing for example, where the median expenditure lies, or what expenditure value lies under the 90th percentile, indicating a top 10% expenditure value.

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Key Concepts

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

Frequency Distribution

Frequency distribution is a method used in data analysis that categorizes data points into groups, known as class intervals, and counts how many data points fall into each group. This way, it transforms raw data into an organized summary.
Imagine you have a collection of numbers, like those representing the 1989 per capita expenditures for public libraries in each of the 50 states. The first step in frequency distribution is to arrange these numbers, either in ascending or descending order. This helps us clearly see the range our data covers, meaning the difference between the highest and lowest values.

• Determine the range of data: Subtract the smallest number in your dataset from the largest number.
• Divide the range by the number of desired class intervals. This calculation will help you decide the size of each class interval.
• Count: Next, for each class interval, you count how many data points fall into it. This number is the frequency for that interval.

The result is a frequency distribution, which provides a clear picture of how your data is spread across its range and makes further analysis, like creating histograms, more feasible.

Histogram

A histogram is a graphical representation of a frequency distribution. It visually shows how your data points are distributed across various intervals.
The horizontal axis (x-axis) of a histogram represents the class intervals. Think of these as the bins into which your data is sorted. On the vertical axis (y-axis), you have the frequencies, which are essentially the number of data points falling into each of the class intervals.

• The height of each bar in the histogram corresponds to the frequency of the class interval it represents.
• The bars are usually adjacent, meaning there's no gap between them, to emphasize the continuous nature of the data.

Histograms are incredibly useful because they make it easy to see patterns within your data at a glance. You can quickly identify things like skewness or the central tendency of the data by looking at the shape and spread of the bars. For example, if most of your bars are clustered on the left side of the histogram, your data might be left-skewed.

Percentile Calculation

Percentile calculation is a statistical method used to interpret and understand the distribution of data. Percentiles divide your data into 100 equal parts. This means that the 90th percentile is the value below which 90% of your data points fall.
To calculate a specific percentile in a dataset:

• Use the formula: \(P_k = \frac{k(N+1)}{100}\), where \(P_k\) is the kth percentile and \(N\) is the total number of observations.
• Calculate \((k(N+1)/100)\) to find the rank of the percentile within your sorted dataset.
• Identify the value in your dataset at this rank to find the percentile value.

Percentiles provide significant insights into your data. They help determine thresholds below which certain percentages of data points lie. This method is especially common in education and exam results to understand relative standing among peers. For instance, being in the 70th percentile on a math test implies that you performed better than 70% of other test-takers. Similarly, calculating percentile values for library expenditures helps identify states with relatively lower or higher spending compared to others.