91Ó°ÊÓ

A sample of 80 adults was taken, and these adults were asked about the number of credit cards they possess. The following table gives the frequency distribution of their responses. $$ \begin{array}{lc} \hline \text { Number of Credit Cards } & \text { Number of Adults } \\ \hline 0 \text { to } 3 & 18 \\ 4 \text { to } 7 & 26 \\ 8 \text { to } 11 & 22 \\ 12 \text { to } 15 & 11 \\ 16 \text { to } 19 & 3 \\ \hline \end{array} $$ a. Find the class boundaries and class midpoints. b. Do all classes have the same width? If so, what is this width? c. Prepare the relative frequency and percentage distribution columns. d. What percentage of these adults possess 8 or more credit cards?

Step by step solution

01

Find the Class Boundaries

Class boundaries are one of the methods used to determine the classes of a frequency distribution. The class boundaries will be one unit less than the lower limit of the next class. So the class boundaries are:\[\begin{array}{lc}\hline \text { Number of Credit Cards } & \text { Class Boundaries } \\hline -0.5 \text { to } 3.5 & 18 \3.5 \text { to } 7.5 & 26 \7.5 \text { to } 11.5 & 22 \11.5 \text { to } 15.5 & 11 \15.5 \text { to } 19.5 & 3 \\hline\end{array}\]

02

Find the Class Midpoints

The class midpoint is the middle value of each class. To get this value, add the upper and lower boundaries of each class and divide by 2. This means \( (lower \ boundary+upper \ boundary)/2 \). Therefore, the class midpoints are : \[ 1.5, 5.5, 9.5, 13.5, 17.5 \]

03

Determine the Class Width

To find the class width, subtract the lower class boundary from the upper class boundary. By checking here, we can see that each class upper boundary minus the class lower boundary is equivalent to 4. Therefore, all the classes have the same width, which is 4.

04

Prepare the Relative Frequency and Percentage Distribution Columns

To calculate the relative frequency of a class, use the formula \(relative \ frequency = frequency/total \ sample \ size \). Ensure that the sum of all relative frequencies equals 1. The percentage distribution can be found by multiplying the relative frequency by 100. Hence, calculate these values for each class accordingly.

05

Calculate Percentage of Adults with 8 or More Credit Cards

To find the percentage of adults with 8 or more credit cards, add the frequencies for the categories 8 to 11, 12 to 15, 16 to 19 and divide by total number of adults then multiply by 100 to convert it to a percentage.

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Key Concepts

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

Class Boundaries

Class boundaries help us smooth out the edges between different classes in a frequency distribution. They serve to separate one group from the next by slightly extending each interval. The boundaries are calculated by subtracting 0.5 from the lower limit and adding 0.5 to the upper limit of each class. This method is useful in avoiding overlaps and ensuring a clear distinction between classes.

For example, the class "0 to 3" becomes "-0.5 to 3.5." This shift is small but significant, as it helps in standardizing the gaps between classes and making frequency distribution more precise.

• Calculating boundaries prevents gaps and overlaps.
• Boundaries provide a smoother representation of data.

Class Midpoints

Class midpoints give us the central value of each class interval, representing the average value of that class. To find the midpoint, add the lower and upper class boundaries and divide the sum by 2. This simple calculation gives a more accurate representation of data within each class.

For the boundary of '-0.5 to 3.5,' the class midpoint is calculated as \( ( -0.5 + 3.5 ) / 2 = 1.5 \). Performing this calculation for each class will yield midpoints such as 1.5, 5.5, 9.5, and so on.

• Midpoints help in identifying the central tendency.
• They are useful for further statistical analysis.

Relative Frequency

Relative frequency portrays how often a particular class occurs as a proportion of the total dataset. It's a crucial element for comparing different intervals effectively. You calculate it by dividing the frequency of each class by the total number of observations:
\( \, \frac{\text{frequency of a class}}{\text{total sample size}} \, \).

Ensuring that all relative frequencies sum up to 1 is essential for consistency. This approach allows easy comparison across classes by using a unified scale.

• Expresses frequency in relation to the total number.
• Helps in understanding the distribution patterns within data.

Percentage Distribution

Percentage distribution converts relative frequencies into an easier-to-understand scale of 100. This makes the data more accessible for interpretation. To find the percentage distribution, multiply each relative frequency by 100.

This conversion allows one to easily visualize how much each class contributes to the entire dataset. For example, a class with a relative frequency of 0.3 would make up 30% of the total data when expressed in percentage terms.

• Makes frequency distributions more intuitive.
• Allows for comparisons over different datasets.