91Ó°ÊÓ

The accompanying table lists the 2006-07 median household incomes (rounded to the nearest dollar), for all 50 states and the District of Columbia. $$ \begin{array}{lccc} \hline \text { State } & \begin{array}{c} \text { 2006-07 Median } \\ \text { Household Income } \end{array} & \text { State } & \begin{array}{c} 2006-07 \text { Median } \\ \text { Household Income } \end{array} \\ \hline \text { AL } & 40,620 & \text { MT } & 42,963 \\ \text { AK } & 60,506 & \text { NE } & 49,342 \\ \text { AZ } & 47,598 & \text { NV } & 53,912 \\ \text { AR } & 39,452 & \text { NH } & 65,652 \\ \text { CA } & 56,311 & \text { NJ } & 65,249 \\ \text { CO } & 59,209 & \text { NM } & 42,760 \\ \text { CT } & 64,158 & \text { NY } & 49,267 \\ \text { DE } & 54,257 & \text { NC } & 42,219 \\ \text { D.C. } & 50,318 & \text { ND } & 44,708 \\ \text { FL } & 46,383 & \text { OH } & 48,151 \\ \text { GA } & 49,692 & \text { OK } & 41,578 \\ \text { HI } & 63,104 & \text { OR } & 49,331 \\ \text { ID } & 48,354 & \text { PA } & 49,145 \\ \text { IL } & 51,279 & \text { RI } & 54,735 \\ \text { IN } & 47,074 & \text { SC } & 42,477 \\ \text { IA } & 49,200 & \text { SD } & 46,567 \\ \text { KS } & 47,671 & \text { TN } & 41,521 \\ \text { KY } & 40,029 & \text { TX } & 45,294 \\ \text { LA } & 39,418 & \text { UT } & 54,853 \\ \text { ME } & 47,415 & \text { VT } & 50,423 \\ \text { MD } & 65,552 & \text { VA } & 58,950 \\ \text { MA } & 57,681 & \text { WA } & 57,178 \\ \text { MI } & 49,699 & \text { WV } & 40,800 \\ \text { MN } & 57,932 & \text { WI } & 52,218 \\ \text { MS } & 36,499 & \text { WY } & 48,560 \\ \text { MO } & 45,924 & & \\ \hline \end{array} $$ a. Construct a frequency distribution table. Use the following classes: \(36,000-40,999,41,000-\) \(45,999,46,000-50,999,51,000-55,999,56,000-60,999,61,000-65,999\) b. Calculate the relative frequencies and percentages for all classes. c. Based on the frequency distribution, can you say whether the data are symmetric or skewed? d. What percentage of these states had a median household income of less than \(\$ 56,000 ?\)

Step by step solution

01

Constructing the Frequency Distribution Table

In this step, data from the table will be sorted into categories. These categories include: $36,000-$40,999; $41,000-$45,999; $46,000-$50,999; $51,000-$55,999; $56,000-$60,999; $61,000-$65,999. For each of these categories, count the number of states that have a median household income within the category's range. Record these as the 'frequency' for each category.

02

Calculating Relative Frequencies and Percentages

To determine the relative frequency for each category, divide the frequency of the category by the total number of states (51 in this case, as it includes District of Columbia). The percentage for each category is found by multiplying the relative frequency by 100.

03

Determination of Symmetry or Skewness

Examining the frequency distribution table, see if the frequencies rise and fall symmetrically (implying a symmetric distribution) or if they rise and fall irregularly or with a tendency towards one side (indicating skewness). If the frequencies are not symmetric and most of the state's incomes are lower than the average income, then the distribution is said to be negatively skewed. While if most incomes are higher than the average it is positively skewed.

04

Percentage of States with Median Household Income less than $56,000

For this, simply sum up the frequencies in all categories with median income less than $56,000 and divide by the total number of states, then multiply by 100 to get the percentage.

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Key Concepts

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

Relative Frequency

Relative frequency is a way to understand how often a particular value appears in a dataset compared to the total number. It's about putting the individual frequency of items in context. When you divide the frequency of a class by the total number of observations, you get the relative frequency.
This gives insight into how common a certain class is relative to the entire dataset.

• Formula: \(\text{Relative Frequency} = \frac{\text{Frequency of a class}}{\text{Total number of observations}}\)
• Used to compare categories or periods.
• Aids in converting raw data into a more understandable form.

By expressing each class as a proportion of the total, it helps us visualize patterns and detect anomalies in data distribution.

Skewness

Skewness tells us about the direction and degree of asymmetry in a data distribution. In frequency distribution, you can determine skewness by looking at how the data clusters around the mean. It helps identify trends in data such as whether most incomes are high or low.

• **Symmetric:** Data is evenly distributed around the mean (bell-shaped curve).
• **Positively skewed:** Tail on the right side is longer and most data points are concentrated on the left.
  This means there are more lower values and a few very high ones.
• **Negatively skewed:** Tail on the left side is longer and most data points are on the right, signaling more high values with a few extremely low ones.

Skewness is useful for understanding the overall shape and balance of data which helps in making predictions and drawing conclusions about trends.

Median Household Income

The median household income represents the middle of a given income distribution. When household incomes are arranged from the lowest to the highest, the median is the middle number, making it a useful statistical measure.
This helps understand typical income levels while being less influenced by extremely high or low incomes than the average.

• Offers a realistic measure of central tendency.
• Less affected by outliers compared to mean.
• Helps in assessing economic conditions and standards of living.

The median provides insight into income equality or inequality within a region, painting a clearer picture of general affluence than individual data points.

Data Categorization

Data categorization involves organizing raw data into groups or classes that make it easier to analyze and interpret. It helps to break down large datasets into smaller, more manageable pieces that offer clear ways to understand complex information.

• Facilitates trend detection and pattern recognition.
• Makes large datasets comprehensible at a glance.
• Enables comparison across different intervals or groups.

By assigning rules or classes for grouping, you can create a frequency distribution that simplifies the data. This is especially useful in practical applications like business analytics, where understanding distributions can influence decision-making and planning.