91Ó°ÊÓ

The following data values are 2009 per capita expenditures on public libraries for each of the \(50 \mathrm{U} . \mathrm{S}\). states (from www.statemaster.com): \(\begin{array}{rrrrrrr}16.84 & 16.17 & 11.74 & 11.11 & 8.65 & 7.69 & 7.48 \\\ 7.03 & 6.20 & 6.20 & 5.95 & 5.72 & 5.61 & 5.47 \\ 5.43 & 5.33 & 4.84 & 4.63 & 4.59 & 4.58 & 3.92 \\ 3.81 & 3.75 & 3.74 & 3.67 & 3.40 & 3.35 & 3.29 \\ 3.18 & 3.16 & 2.91 & 2.78 & 2.61 & 2.58 & 2.45 \\ 2.30 & 2.19 & 2.06 & 1.78 & 1.54 & 1.31 & 1.26 \\ 1.20 & 1.19 & 1.09 & 0.70 & 0.66 & 0.54 & 0.49 \\ 0.30 & 0.01 & & & & & \end{array}\) 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. \(90 \mathrm{th}\) ii. 70 th v. \(40 \mathrm{th}\) iii. 10 th

Step by step solution

01

Sort Data and Calculate Frequency

Firstly, make sure to sort the data from smallest to largest. Then count up frequencies of values. The frequency indicates how often a particular value appears. As there are many unique values, it may be useful to choose some interval class (say, having an interval class of 1.0, for example).

02

Construct Histogram

Now, with the sorted values and corresponding frequencies, a histogram can be constructed. The horizontal axis will present the intervals of the data and vertical axis will represent the frequency. The bar height corresponds to the frequency of the values within the particular interval.

03

Calculate Percentiles

The next step is to calculate percentiles. Percentiles are measures that divide the data into 100 equal parts. The nth percentile corresponds to the value below which n percent of the data falls. To calculate the 50th, 70th, 10th, 90th, and 40th percentiles, locate the equivalent area on the histogram first. For example, to find the 50th percentile, find the value which splits off the lowest 50% of data from the highest 50%. In other words, it's the median of the data.

04

Recap and check for understanding

Summarize all the steps and double check that all parts of the exercise have been fulfilled: the frequency distribution has been calculated and presented, the histogram has been created, and the percentiles have been approximately valued.

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Key Concepts

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

Frequency Distribution

Understanding frequency distribution is essential when analyzing sets of data such as expenditures on public libraries. A frequency distribution is simply an organized tally of how often each value in a set of data occurs. In the given problem, we might decide to group the expenditures into ranges (also known as bins) to make it easier to see patterns. For instance, one possible grouping could be in increments of \(1.00—how many states spend between \)0.00–\(1.00, \)1.00–$2.00, and so forth. By counting the number of observations within each of these ranges, you produce a frequency table that can then be visualized as a histogram. This process transforms raw data into a form where trends might be more easily spotted.

Histogram

A histogram provides a visual representation of frequency distribution. It consists of bars where each bar's height is proportional to the frequency of values within the interval it represents. When constructing a histogram from our expenditure data, the horizontal axis (the x-axis) would depict the range of expenditures, divided into bins, while the vertical axis (the y-axis) would represent the frequency of states within each bin. For the provided data, this bar chart will help students see which expenditure ranges are most common among the states and identify any outliers at a glance. It's a powerful tool for summarizing large data sets and can help in understanding the overall distribution pattern, like whether it's skewed toward higher or lower expenditure values.

Percentiles

Percentiles are a way to comprehend the relative standing of a value within a data set. The nth percentile tells you what percentage of the data falls below that value. For instance, the 50th percentile, also known as the median, is the middle value where half the data is lower and the other half is higher. On the histogram, you would find the 50th percentile by looking for the value that divides the area under the histogram into two equal parts. Similarly, the 10th percentile separates the lowest 10% of the data from the rest, and so forth. This method of dividing the data into one hundred equal parts can provide insight into the distribution's spread and center, and is especially useful when comparing different data sets or examining the distribution's shape.

Data Analysis

Analyzing data involves collecting, cleaning, interpreting, and presenting it. After creating a frequency distribution and a histogram, we proceed to examine the data for patterns, relationships, and insights. In the context of public library expenditures, we would analyze the histogram and calculate various percentiles to understand not just the average or mid-range spending, but also how spending is distributed across states. By doing this, we can identify which states are spending significantly more or less than others, which spending ranges are most common, or which percent of states fall below a certain expenditure threshold. This statistical analysis helps to make informed decisions and strategies in policy-making, budget allocation, and understanding public investment trends.