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Chapter 6: Problem 27

An advertising company plans to market a product to low-income families. A study states that for a particular area, the average income per family is \(\$ 24,596\) and the standard deviation is \(\$ 6256 .\) If the company plans to target the bottom \(18 \%\) of the families based on income, find the cutoff income. Assume the variable is normally distributed.

Short Answer

Expert verified
The cutoff income is approximately $18,871.

Step by step solution

01

Understand the Problem

We need to find the cutoff income that corresponds to the bottom 18% of family incomes, given that incomes are normally distributed. The average (mean) income is $24,596 and the standard deviation is $6,256.
02

Identify the Z-score

Since we are dealing with a normal distribution, we use a Z-score table or calculator to find the Z-score corresponding to the bottom 18%. The Z-score for the 18th percentile is approximately -0.915.
03

Use the Z-score Formula

Apply the formula to convert the Z-score to an actual income value:\[ X =  ext{mean} + Z imes ext{standard deviation} \]where \( X \) is the cutoff income, the mean is \(24,596, and the standard deviation is \)6,256. Substitute the values to get:\[ X = 24,596 + (-0.915) imes 6,256 \]
04

Calculate the Cutoff Income

Calculate the expression from the previous step:1. Multiply the Z-score by the standard deviation: \(-0.915 \times 6,256 = -5,725.04\).2. Add this result to the mean: \( 24,596 - 5,725.04 = 18,870.96 \).Therefore, the cutoff income is approximately $18,871.

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Key Concepts

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

Statistical Analysis
In simple terms, statistical analysis is a way to collect, review, and interpret data. It helps us understand how different variables interact with each other.
This is important when studying something like income levels in a community. By analyzing data statistically, we can derive meaningful insights that are not immediately obvious.
  • Allows for better decision-making by providing a clearer picture of the data.
  • Can be used to predict future trends by looking at past data patterns.
  • Helps identify relationships between different datasets, enabling more precise targeting for initiatives like marketing campaigns.
Statistical analysis is the foundation for many other concepts such as the percentile rank and z-score calculation mentioned below.
Percentile Rank
The percentile rank is a measure that tells you the percentage of values below a particular value in a data set. In our context, the percentile rank is used to identify income levels of specific groups. For example, the 18th percentile refers to the income level that separates the bottom 18% of families from the rest.
Understanding percentile ranks is crucial when targeting specific segments like low-income families. It helps you pinpoint exactly where your target group falls within the overall distribution.
  • It's a relative measure, meaning it doesn't express the value in concrete terms like dollars, but as a position within the data.
  • Useful in fields such as education, finance, and marketing, where understanding group distributions can impact decisions and strategies.
Z-score Calculation
A Z-score calculation helps in converting individual data points into a common metric, showing how far a point is from the mean. It's a score that indicates how many standard deviations an element is from the mean.
In the context of a normal distribution, a particular Z-score corresponds to a particular percentile. For example, a Z-score of -0.915 corresponds to the 18th percentile.
  • The Z-score formula is: \[ Z = \frac{(X - \text{mean})}{\text{standard deviation}} \]where \( X \) is the value of interest.
  • Helps to easily find probabilities and cutoffs in a normal distribution.
  • Helps in determining if a value is typical or atypical within a data set.
This is why Z-scores are such a handy tool in statistical analysis.
Mean and Standard Deviation
Understanding mean and standard deviation is key when analyzing data distributions. The mean is simply the average of a data set. It provides a central value toward which all other data points gravitate.
Standard deviation, on the other hand, measures how spread out the data points are around the mean. A higher standard deviation means more spread, while a lower one means the data points are closer to the mean.
  • The formula for the mean is: \[ \text{Mean} = \frac{\Sigma X}{N} \]where \( \Sigma X \) is the sum of all data points and \( N \) is the number of data points.
  • Standard deviation is calculated as: \[\text{Standard deviation} = \sqrt{\frac{\Sigma (X_i - \text{mean})^2}{N}}\]
  • Both these metrics help in understanding the overall shape and extent of a distribution.
They are foundational for any further analysis, such as when calculating cutoffs using Z-scores.
Income Distribution Analysis
Income distribution analysis focuses on studying how income is shared among different participants in an economy. It often uses statistical tools like those mentioned above to draw conclusions.
In our example, the purpose of analyzing income distribution is to identify the bottom 18% of income earners for targeted marketing.
  • Identifying specific income segments allows for more tailored and effective marketing strategies.
  • It uncovers underlying trends, such as income inequality, that can guide policy and strategy adjustments.
  • Informs businesses and policymakers about the economic conditions that could affect their decisions.
Income distribution analysis helps you see the bigger picture of economic landscapes and make informed decisions.

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Most popular questions from this chapter

To qualify for a police academy, applicants are given a test of physical fitness. The scores are normally distributed with a mean of 64 and a standard deviation of 9. If only the top 20% of the applicants are selected, find the cutoff score.

The average annual salary for all U.S. teachers is \(\$ 47,750\). Assume that the distribution is normal and the standard deviation is \(\$ 5680 .\) Find the probability that a randomly selected teacher earns a. Between \(\$ 35,000\) and \(\$ 45,000\) a year b. More than \(\$ 40,000\) a year c. If you were applying for a teaching position and were offered \(\$ 31,000\) a year, how would you feel (based on this information)?

The average daily jail population in the United States is 706,242. If the distribution is normal and the standard deviation is 52,145, find the probability that on a randomly selected day, the jail population is a. Greater than 750,000 b. Between 600,000 and 700,000

According to the World Almanac, 72% of households own smartphones. If a random sample of 180 households is selected, what is the probability that more than 115 but fewer than 125 have a smartphone?

Find the probabilities for each, using the standard normal distribution. \(P(-2.07< z<1.88)\)
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