You Are A Marketing Manager For A Very Important Shopping Ce

You Are A Marketing Manager For A Very Important Shopping Center In Yo

You are a marketing manager for a very important shopping center in your city. You are interested in studying the behavior of buyers with respect to the amount spent and the time they make their purchases in order to propose a new advertising campaign that also involves granting certain discounts and promotions. It has been concluded from the study that 70% of the shoppers in the shopping center make their purchases on weekends and also spend more money during that period. If the amount of money buyers spend on Saturdays between 5 pm and 8 pm has a normal distribution with mean $90 and standard deviation of $5. Then you select a buyer at random and want to determine how they will spend their money.

Paper For Above instruction

This analysis aims to evaluate consumer spending behavior within a specific period at a prominent shopping center to inform targeted marketing strategies. Recognizing that a significant portion of shoppers, specifically 70%, make purchases during weekends and tend to spend more, this study focuses on Saturday evening transactions between 5 pm and 8 pm, where expenditure follows a normal distribution with a mean of $90 and a standard deviation of $5. Understanding this distribution allows us to compute probabilities related to spending thresholds, which can guide promotional activities, discounts, and advertising campaigns to optimize sales and customer engagement.

First, it is essential to understand the properties of the normal distribution in this context. Given that the amount spent (X) on Saturday evenings is normally distributed with parameters μ = $90 and σ = $5, we can compute probabilities using standard statistical methods. The total area under the normal curve represents the probability of outcomes within specific spending ranges, enabling marketers to predict customer behavior effectively.

1. Probability of spending more than $90 and less than $90

To determine the probability that a randomly selected buyer spends more than $90, we use the symmetry property of the normal distribution. Since the mean is $90, the probability that a buyer spends more than $90 is exactly 0.5, or 50%. Similarly, the probability of spending less than $90 is also 50%. Mathematically:

• P(X > $90) = 0.5
• P(X

This is because $90 is the mean of the distribution, and the normal distribution is symmetric about the mean.

2. Probability that expenditure lies between $80 and $100

Next, to find the probability that a buyer spends between $80 and $100, we calculate the standardized z-scores for these values:

• z for $80: (80 - 90) / 5 = -2
• z for $100: (100 - 90) / 5 = 2

Using standard normal distribution tables or a calculator, the probabilities corresponding to these z-scores are:

• P(Z
• P(Z

Therefore, the probability that the expenditure falls between $80 and $100 is:

P($80

Approximately 95.44% of shoppers spend between $80 and $100 during Saturday evenings in this period, indicating a high concentration around the mean, which validates previous assumptions about typical spending behavior.

3. Standardize the probability distribution

The standardization process converts the original normal distribution into the standard normal distribution (Z), which has a mean of 0 and a standard deviation of 1. The standardized z-score for any value X is calculated as:

z = (X - μ) / σ

For example, for X = $90, the z-score is:

z = (90 - 90) / 5 = 0

This transformation allows us to analyze probabilities across different ranges by referencing standard normal tables or computational tools, facilitating easier calculation of likelihoods for various spending amounts.

4. Probability of spending less than $50 using the standardized function

Now, to find the probability that a buyer spends less than $50, we compute the z-score:

z = (50 - 90) / 5 = -40 / 5 = -8

Consulting standard normal distribution tables or using computational tools, the probability associated with z = -8 is extremely close to zero, effectively zero for practical purposes. Consequently:

P(X

This indicates that the likelihood of a buyer spending less than $50 on Saturday evening in this context is virtually nonexistent, underlining the concentration of spending around the mean value.

5. Graphical representation

The probability distribution can be visually represented by a bell-shaped curve centered at $90, extending symmetrically with the specified standard deviation. The shaded areas under the curve between -2 and 2 z-scores (corresponding to $80 and $100) depict the 95.44% probability of spending within this range. Furthermore, a tiny area far to the left (beyond -8 standard deviations) highlights the negligible probability of expenditures below $50. Such visualizations aid in comprehending the distribution's characteristics and the likelihood of various spending outcomes.

Graphically, the normal distribution would show the highest point at $90, with rapid thinning tails, emphasizing that most spending occurs near the mean. Marking the $50 point well into the left tail demonstrates its rarity, essential information when designing targeted marketing promotions.

Conclusion

Understanding the normal distribution of buyer expenditures enables effective segmentation and targeting in marketing campaigns. Recognizing that most consumers spend between $80 and $100, with only negligible chances of significantly lower expenditure, guides the formulation of discounts and promotional strategies tailored to typical customer behavior. Accurate probability calculations and graphical representations provide vital insights, enhancing the shopping center’s ability to optimize sales during peak periods.

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