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What is the age distribution of promotion-sensitive shoppers? A supermarket super shopper is defined as a shopper for whom at least \(70 \%\) of the items purchased were on sale or purchased with a coupon. The following table is based on information taken from Trends in the United States (Food Marketing Institute, Washington, D.C.). $$\begin{array}{|l|ccccc|} \hline \text { Age range, years } & 18-28 & 29-39 & 40-50 & 51-61 & 62 \text { and over } \\\ \hline \text { Midpoint } x & 23 & 34 & 45 & 56 & 67 \\ \hline \begin{array}{l} \text { Percent of } \\ \text { super shoppers } \end{array} & 7 \% & 44 \% & 24 \% & 14 \% & 11 \% \\ \hline \end{array}$$ For the 62 -and-over group, use the midpoint 67 years. (a) Using the age midpoints \(x\) and the percentage of super shoppers, do we have a valid probability distribution? Explain. (b) Use a histogram to graph the probability distribution of part (a). (c) Compute the expected age \(\mu\) of a super shopper. (d) Compute the standard deviation \(\sigma\) for ages of super shoppers.

Step by step solution

01

Check Validity of Probability Distribution

To determine if the given distribution is a valid probability distribution, ensure that the sum of all the percentages adds up to 100%. Given percentages are: 7%, 44%, 24%, 14%, 11%. Sum = 7% + 44% + 24% + 14% + 11% = 100%. Thus, it is a valid probability distribution.

02

Histogram Construction

To graph the probability distribution, plot age midpoints on the x-axis and their corresponding percentages on the y-axis. 1. Create a bar for each midpoint: 23, 34, 45, 56, and 67. 2. The height of each bar should match the percentage of super shoppers in each age range: 7, 44, 24, 14, and 11, respectively.

03

Calculate Expected Age (Mean)

The expected value (mean) is computed as:\[ \mu = \sum (x \times P(x)) \]Where:- \(x\) is the midpoint for each age range.- \(P(x)\) is the probability for each age range (percentage divided by 100).Calculations:\[ \mu = (23 \times 0.07) + (34 \times 0.44) + (45 \times 0.24) + (56 \times 0.14) + (67 \times 0.11) \]\[ \mu = 1.61 + 14.96 + 10.8 + 7.84 + 7.37 \]\[ \mu = 42.58 \]The expected age \(\mu\) is approximately \(42.58\) years.

04

Calculate Standard Deviation

First, compute the variance using:\[ \sigma^2 = \sum (x^2 \times P(x)) - \mu^2 \]Compute \(x^2 \times P(x)\) for each category:1. \(23^2 \times 0.07 = 37.03\)2. \(34^2 \times 0.44 = 510.24\)3. \(45^2 \times 0.24 = 486\)4. \(56^2 \times 0.14 = 439.04\)5. \(67^2 \times 0.11 = 493.69\)Sum these values:\[ \sum = 37.03 + 510.24 + 486 + 439.04 + 493.69 = 1965.99\]Subtract the square of the mean:\[ \sigma^2 = 1965.99 - 42.58^2 \]\[ \sigma^2 = 1965.99 - 1812.2564 = 153.7336 \]Then, compute the standard deviation:\[ \sigma = \sqrt{153.7336} \approx 12.4 \]The standard deviation \(\sigma\) is approximately \(12.4\) years.

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Key Concepts

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

Histogram

A histogram is a common way to visualize the distribution of data. It uses bars to represent the frequency or probability of different ranges of data. In the context of our problem, we use a histogram to show how different age groups of promotion-sensitive super shoppers are distributed.
To create a histogram:

• Place the age midpoints on the x-axis at intervals like 23, 34, 45, 56, and 67 years.
• The height of each bar should reflect the percentage of super shoppers in each age group: 7%, 44%, 24%, 14%, and 11%, respectively.
• Each bar’s height visually represents how likely a shopper belongs to a particular age range when chosen at random.

This visual aid helps to easily understand which age group comprises the largest segment of these promotion-sensitive shoppers. You'll often see data concentrated in certain bins, and this can show trends, gaps, or unusual observations in data.

Expected Value

The expected value, often represented by the symbol \( \mu \), is a fundamental concept in probability and statistics. It provides the average value you'd expect from a random scenario. For our super shopper age distribution:

• The expected value is calculated as a weighted average of all possible age midpoints, where each midpoint is weighted by the probability (the percentage divided by 100) of that age group.
• Using the formula: \[ \mu = \sum (x imes P(x)) \] where \( x \) is each midpoint value, and \( P(x) \) is the corresponding probability.

For our data set, the expected age is approximately 42.58 years. It implies that the average age of a promotion-sensitive super shopper is around this number. The expected value gives a sense of the center of the distribution, but it doesn't reflect the spread or variability of the data.

Standard Deviation

The standard deviation \( \sigma \) indicates how much the ages of promotion-sensitive super shoppers differ from the expected value. It's a measure of the dispersion or variability in a data set.

The steps to calculate the standard deviation include:

• First, find the variance, which is the average of the squared differences from the mean.
• Using the formula: \[ \sigma^2 = \sum (x^2 \times P(x)) - \mu^2 \] where \( x \) are the midpoints and \( P(x) \) are the probabilities.
• After computing the variance, take the square root to get the standard deviation: \[ \sigma = \sqrt{variance} \]

In our example, the standard deviation works out to approximately 12.4 years. This means there's a significant spread in the ages of these super shoppers around the expected average age of 42.58 years. Knowing the standard deviation helps us understand how diverse our data is.

Variance

Variance is a crucial statistical concept that tells us about the spread of a data set. It's the mean of the squares of the deviations (or differences) from the expected value.

In our context, variance provides insight into the distribution of ages among promotion-sensitive super shoppers.

• The calculation involves squaring each age group's deviation from the mean, weighted by its probability, then averaged to get the variance.
• Mathematically, variance is expressed as: \[ \sigma^2 = \sum (x^2 \times P(x)) - \mu^2 \] where \( x \) is each midpoint, \( P(x) \) is the specific probability, and \( \mu \) is the mean.

In our problem, the variance was calculated to be approximately 153.73. This value reflects a considerable variability in the ages of the super shoppers from the expected age. Variance is pivotal as it highlights how much the ages deviate, but to interpret it more intuitively, we often take its square root to derive the standard deviation.