Market Research Indicates Customers Are

Market Research Has Indicated That Customers Are

Market research has indicated that customers are likely to bypass Roma tomatoes that weigh less than 70 grams. A produce company produces Roma tomatoes that average 74.0 grams with a standard deviation of 3.2 grams. (a) Assuming that the normal distribution is a reasonable model for the weights of these tomatoes, what proportion of Roma tomatoes are currently undersize (less than 70g)? (b) How much must a Roma tomato weigh to be among the heaviest 10%? (c) The aim of the current research is to reduce the proportion of undersized tomatoes to no more than 2%. One way of reducing this proportion is to reduce the standard deviation. If the average size of the tomatoes remains 74.0 grams, what must the target standard deviation be to achieve the 2% goal? (d) The company claims that the goal of 2% undersized tomatoes is reached. To test this, a random sample of 25 tomatoes is taken. What is the distribution of undersized tomatoes in this sample if the company's claim is true? Explain your reasoning.

Paper For Above instruction

Market research indicating consumer preferences has a significant impact on product sizing and production strategies in the agricultural sector. Specifically, in the case of Roma tomatoes, understanding weight distributions allows producers to meet quality standards, minimize waste, and align with consumer expectations. This paper discusses the application of normal distribution assumptions to analyze tomato weights, determine proportions of undersized produce, evaluate thresholds for the heaviest tomatoes, and explore how standard deviation adjustments can achieve targeted quality metrics.

Analyzing the Distribution and Proportions of Tomato Weights

The first step involves calculating the proportion of Roma tomatoes that weigh less than 70 grams. Assuming the weights follow a normal distribution with mean μ = 74.0 grams and standard deviation σ = 3.2 grams, we utilize the standard normal distribution (z-score) for this calculation. The z-score corresponding to 70 grams is:

z = (X - μ) / σ = (70 - 74.0) / 3.2 = -4.0 / 3.2 ≈ -1.25.

The cumulative probability for z = -1.25, from standard normal distribution tables or computational tools, is approximately 0.1056. Therefore, roughly 10.56% of the Roma tomatoes are undersized (

Determining the Heaviest 10% of Roma Tomatoes

To identify the weight cutoff for the top 10% of tomatoes (the heaviest), we find the 90th percentile of the distribution. The z-score for the 90th percentile is approximately 1.28. The corresponding weight (X) is given by:

X = μ + z σ = 74 + 1.28 3.2 ≈ 74 + 4.096 ≈ 78.096 grams.

Thus, a Roma tomato must weigh at least approximately 78.10 grams to be among the heaviest 10%, which can guide marketing and packaging decisions for premium products.

Reducing the Proportion of Undersized Tomatoes to 2%

The goal is to adjust the production process so that only 2% of tomatoes fall below 70 grams. Maintaining the mean at 74.0 grams, we need to determine the standard deviation σ' that achieves this target. We find the z-score corresponding to the 2nd percentile, which is approximately -2.05. Re-arranging the z-score formula:

σ' = (X - μ) / z = (70 - 74) / -2.05 ≈ (-4) / -2.05 ≈ 1.95 grams.

Therefore, to reduce the undersizing proportion to 2%, the standard deviation should be reduced to approximately 1.95 grams, indicating a need for tighter quality control in the growing or harvesting process.

Sampling Distribution of Undersized Tomatoes in a New Sample

If the company's goal of 2% undersized tomatoes is achieved through process modification, and the claim is correct, then in a sample of 25 tomatoes, the number of undersized tomatoes follows a binomial distribution with parameters n = 25 and p = 0.02. The expected number of undersized tomatoes would be:

E = n p = 25 0.02 = 0.5.

The variance of this distribution is:

Var = n p (1 - p) = 25 0.02 0.98 ≈ 0.49.

The distribution of undersized tomatoes in the sample approximates a binomial distribution with these parameters. Since p is small and n is moderate, a Poisson approximation with λ = 0.5 could be used for simplicity, implying that in most samples, either zero or one tomato would be undersized, confirming the effectiveness of the process adjustment.

This distribution reflects the company's claim that the process can reliably produce a negligible proportion of undersized tomatoes.

Conclusion

Applying normal distribution assumptions in agricultural quality control enables producers to evaluate current standard deviations, set realistic thresholds for premium products, and improve manufacturing processes. Achieving a lower proportion of undersized tomatoes through reduction in variability demonstrates the importance of precise cultivation, harvesting, and sorting techniques. Regular sampling and statistical analysis ensure adherence to quality standards and support strategic decision-making in produce industry practices.

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