SOC212 - Application Question 2 Due April 8 At 11

SOC212 - Application Question #2 Due Friday, April 8th at 11:59pm How different is different enough?

The assignment involves analyzing whether a new, taste-free, fat-free potato-chip alternative is worth market investment. Users rated these chips on a scale from 1 to 10, with the average rating being 5.1 based on 500 tasters. Your task is to evaluate if marketing this product for $10 million is justified based on the ratings, using statistical hypothesis testing methods, specifically a z-test. You should discuss and explain how you arrive at your conclusion, considering the normal distribution, probability, and significance levels.

Paper For Above instruction

The advent of innovative food products continuously challenges conventional market strategies, especially when these products promise to revolutionize consumption habits through technological or compositional breakthroughs. One such hypothetical innovation is a potato chip devoid of potatoes, with an indefinite shelf life, and free of both fat and taste. Despite the novelty, consumer acceptance hinges on subjective taste ratings, which are critical in assessing whether the product warrants substantial market investment, such as a $10 million expenditure. This paper explores whether the new chip's average taste rating of 5.1 justifies its commercialization, using inferential statistics and hypothesis testing principles grounded in the normal distribution.

Hypothesis testing is a key statistical tool used to make informed decisions based on sample data. In this context, the question is whether the average rating of 5.1 is significantly different from a relevant benchmark or neutral point on the rating scale, which is presumed to be 5, representing an indifferent taste perception. The null hypothesis (H₀) posits that there is no difference in taste perception; that is, the true mean rating is 5. Conversely, the alternative hypothesis (H₁) suggests that the true mean rating differs from 5, and in this case, we examine whether the rating is significantly lower or higher. Given the centrality of consumer taste ratings in determining market success, establishing statistical significance beyond random variation is essential.

The statistical framework employs a z-test for a single sample mean, appropriate due to the large sample size (n=500), which ensures the sampling distribution of the mean approaches normality according to the Central Limit Theorem. The test calculates a z-score, which indicates how many standard errors the sample mean is from the hypothesized population mean (here, 5). The formula for the z-score is:

z = (X̄ - μ) / (σ / √n)

where X̄ is the sample mean (5.1), μ is the hypothesized mean (5), σ is the population standard deviation (which should be known or estimated), and n is the sample size. Given the large sample, the standard error (SE) is computed as σ / √n. Without precise notation for σ, one can infer from prior ratings or assume a standard deviation consistent with taste ratings, typically around 1 point, but for rigorous analysis, this parameter must be estimated or obtained.

Calculating the z-score allows us to determine the likelihood of observing such a sample mean if the null hypothesis is true. We compare the obtained z-value to critical z-values for a chosen significance level (α), commonly 0.05 (5%). If the z-score exceeds the critical value in magnitude (either positive or negative), we reject the null hypothesis, concluding that the difference in taste ratings is statistically significant.

In this scenario, the computed z-score might be close to (5.1 - 5) / standard error, and if the resulting p-value associated with this z exceeds 0.05, the difference is not statistically significant, suggesting that the taste perception, on average, is effectively neutral or acceptable to consumers. Conversely, if the difference is statistically significant and indicates a lower rating, it might point to consumer dissatisfaction, thus risking the product’s market success.

Furthermore, understanding the area under the normal curve corresponding to the calculated z-score informs the probability of such a rating occurring due to chance. If the p-value is sufficiently small, it argues against chance variation, supporting the conclusion that the rating is either genuinely high or low. This statistical evidence guides whether the $10 million investment in marketing is justified—if the taste rating is statistically acceptable or better than the threshold, marketing might proceed; if not, reconsideration is warranted.

In conclusion, conducting a z-test to evaluate the mean taste rating provides an objective measure of consumer acceptance, allowing decision-makers to weigh the risk and potential return of marketing the novel chip. The approach underscores the importance of statistical inference in business decision-making, where small differences can be the difference between success and failure.

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