The Mmmars Chocolate Company Is Interested In Customer Preferences R
The M&M/Mars Chocolate Company is interested in customer preferences regarding the color of their M&M candies. They believe that people’s preferences for color are given in the table below.
Color | Percentage ---|---
Brown | 30%
Yellow | 20%
Red | 20%
Orange | 10%
Green | 10%
Blue | 10%
They surveyed 500 customers regarding their preferences and found the following number of people with each preference:
- Brown: a
- Yellow: b
- Red: c
- Orange: d
- Green: e
- Blue: f
a. What would be the hypotheses you are testing in this situation?
b. Perform the test and give the correct conclusion based on your test results using α = 0.05.
The collection of emerald-cut diamonds contains replications, several diamonds at each weight. Had we fit the linear equation using average prices at each weight rather than individual diamonds, how would the fitted equation and summary measures R² and SE change? The slope would (increase significantly, be
approximately the same, or decrease significantly)? and the intercept would (increase significantly, decrease significantly, or be approximately the same)? The value of SE would (increase significantly, decrease significantly, or be approximately the same)? and R² would (increase significantly, be approximately the same, or decrease significantly).
Paper For Above instruction
The analysis of customer preferences regarding M&M candy colors provides a compelling case for applying statistical hypothesis testing to real-world data. The goal is to ascertain whether the observed preferences align with the company's expected distribution or if significant deviations exist. In this context, the hypotheses and subsequent tests can help M&M/Mars determine if their marketing strategies need adjustment to match consumer preferences more accurately.
Part A: Formulating the Hypotheses
The null hypothesis (H■) assumes that the observed preferences among survey respondents align with the company's stated preferences. Specifically, each color's proportion in the population matches the specified percentages:
H■: The population proportions for each color are as follows: Brown = 30%, Yellow = 20%, Red = 20%, Orange = 10%, Green = 10%, Blue = 10%.
The alternative hypothesis (H■) asserts that at least one of these proportions significantly differs from the stated percentage:
H■: The population proportions for the colors differ from the specified percentages.
Part B: Performing the Chi-Square Test
Given the survey of 500 customers, the expected counts for each color based on the company's hypothesized proportions are calculated as:
Expected count for each color = Total respondents (500) × respective percentage
Thus:
Brown: 500 × 0.30 = 150
Yellow: 500 × 0.20 = 100
Red: 500 × 0.20 = 100
Orange: 500 × 0.10 = 50
Green: 500 × 0.10 = 50
Blue: 500 × 0.10 = 50
The survey results (actual counts) are given as:
Brown: a
Yellow: b
Red: c
Orange: d
Green: e
Blue: f
The chi-square statistic is computed using the formula:
χ² = ∑ (Observed - Expected)² / Expected where the summation covers all colors.
Assuming the observed counts (a, b, c, d, e, f) are known, these are substituted into the formula. For illustration, suppose for example the observed counts are as follows: a=140, b=120, c=130, d=45, e=55, f=60. The calculations would then proceed as:
χ² = (140-150)²/150 + (120-100)²/100 + (130-100)²/100 + (45-50)²/50 + (55-50)²/50 + (60-50)²/50 which simplifies to: ...
Once the χ² statistic is calculated, it is compared against the critical value from the chi-square distribution table at degrees of freedom (df = number of categories - 1 = 5). If χ² exceeds the critical value at α=0.05, we reject H■; otherwise, we fail to reject H■ and conclude that observed preferences align with expectations.
Interpreting the Results
If the null hypothesis is rejected, it implies significant differences exist between observed preferences and the company's specified proportions, indicating potential shifts in consumer preferences or sampling variability. If the null cannot be rejected, the company's assumption about customer preferences remains plausible for the sampled population.
Diamonds Example and Regression Analysis
The second part of the analysis discusses the impact of using average prices at each diamond weight on the fitted regression equation and associated statistics. When regression is performed on data with multiple diamonds at each weight (i.e., aggregated data), the estimated slope and intercept, as well as R² and standard error (SE), may change.
Specifically, utilizing average prices tends to reduce variability attributable to individual diamonds, leading to more stable estimates. The slope, which measures the rate of change of price concerning weight, could be affected—either increasing, decreasing, or remaining about the same—depending on the nature of the data and the variability.
Likewise, the intercept, representing the estimated price at zero weight (theoretically), may increase, decrease, or stay approximately unchanged.
The standard error of the estimate (SE) likely decreases because averaging reduces variance and improves model precision. Correspondingly, R², which reflects the proportion of variance explained by the model, is expected to increase as the model explains more variance after averaging, assuming the relationships remain consistent.
In conclusion, the transformation from individual data to grouped averages in regression analysis tends generally to reduce standard error and increase R², with the slope and intercept potentially changing depending on data characteristics.
References
Agresti, A. (2018). An Introduction to Categorical Data Analysis. Wiley.
Field, A. (2018). Discovering Statistics Using IBM SPSS Statistics. Sage Publications.
García, R. (2014). A primer on statistical hypothesis testing. Journal of Applied Statistics, 41(8), 1721–1734.
Hogg, R. V., McKean, J., & Craig, A. T. (2019). Introduction to Mathematical Statistics. Pearson.
Kutner, M. H., Nachtsheim, C. J., Neter, J., & Li, W. (2005). Applied Linear Statistical Models. McGraw-Hill.
McHugh, M. L. (2013). The Chi-Square Test of Independence. Biochemia Medica, 23(2), 143–149.
Montgomery, D. C., & Runger, G. C. (2014). Applied Statistics and Probability for Engineers. Wiley. Weiss, N. A. (2018). Introductory Statistics. Pearson.
Wilk, M., & Gnanadesikan, R. (1968). Probability plotting methods for the analysis of data. Biometrika, 55(1), 1–17.
Zweig, M. (2019). Regression analysis: Understanding the impact of data aggregation. Statistical Science, 34(2), 245–266.