Lingerie Style Preference: Sign Test Analysis

From Two Styles to a Tournament

When you have three or more styles to compare, such as a 'Plunge' bra, a 'Balconette', and a 'Wireless Bralette', you cannot use a single sign test. Instead, the most direct approach is to conduct a series of pairwise comparisons for every possible combination:

• Plunge vs. Balconette
• Plunge vs. Wireless
• Balconette vs. Wireless

This creates a small tournament, where each style is tested against every other.

For each of these individual pairings, the process is exactly what you learned in the last lesson. You collect preference data, discard ties, and perform a sign test.

Once all the tests are complete, the final step—and the focus of today's lesson—is to synthesize the results. Do the individual outcomes point to a clear winner? Is there a clear loser? Or are the preferences circular, with no single best option?

Numerical Exercise: Crowning a Winner

You are in the final selection phase for a photoshoot and need to choose the best bra style from three options for your model. The candidates are:

• Style A: Plunge Bra
• Style B: Balconette Bra
• Style C: Wireless Bralette

You conduct a series of pairwise tryouts, pairing each style with various outfits. You are interested in finding if there is any significant preference among them, so you will use two-sided tests. After discarding ties, you are left with the following preference counts:

1. Plunge (A) vs. Balconette (B):

   • Total non-tied trials: $n=18$
   • Model preferred Plunge: 14 times
   • Model preferred Balconette: 4 times

2. Plunge (A) vs. Wireless (C):

   • Total non-tied trials: $n=16$
   • Model preferred Plunge: 12 times
   • Model preferred Wireless: 4 times

3. Balconette (B) vs. Wireless (C):

   • Total non-tied trials: $n=15$
   • Model preferred Balconette: 4 times
   • Model preferred Wireless: 11 times

Your Task:
For each of the three pairings, perform a two-sided sign test at a significance level of $\alpha =0.05$. Then, based on the collective results, determine which style, if any, is the winner.

• State the null and alternative hypotheses for each test.
• Calculate the two-sided p-value for each test. Use the individual binomial probabilities provided below. Remember, for a two-sided test where the observed count $k$ is in the upper tail, the p-value is $2\times P(X\ge k)$.
• Conclude for each test whether you reject or fail to reject the null hypothesis.
• Synthesize the three results to provide a final recommendation.

Binomial Probabilities for $p=0.5$:

• For n=18:
  • $P(X=14)\approx 0.0117$
  • $P(X=15)\approx 0.0036$
  • $P(X=16)\approx 0.0008$
  • $P(X=17)\approx 0.0001$
  • $P(X=18)\approx 0.0000$
• For n=16:
  • $P(X=12)\approx 0.0278$
  • $P(X=13)\approx 0.0085$
  • $P(X=14)\approx 0.0018$
  • $P(X=15)\approx 0.0002$
  • $P(X=16)\approx 0.0000$
• For n=15:
  • $P(X=11)\approx 0.0417$
  • $P(X=12)\approx 0.0139$
  • $P(X=13)\approx 0.0032$
  • $P(X=14)\approx 0.0005$
  • $P(X=15)\approx 0.0000$

Solution and Interpretation

Let's break down the analysis one test at a time.

Test 1: Plunge (A) vs. Balconette (B)

• Hypotheses:
  • $H_0:p=0.5$ (No preference between Plunge and Balconette)
  • $H_1:p\ne 0.5$ (A significant preference exists)
• Data: $n=18$, with $k=14$ preferences for the Plunge bra.
• p-value Calculation:
  The observed count is $k=14$. We calculate $P(X\ge 14)$ and double it. $$P(X\ge 14)=P(14)+P(15)+P(16)+P(17)+P(18)$$ $$P(X\ge 14)\approx 0.0117+0.0036+0.0008+0.0001+0.0000=0.0162$$ The two-sided p-value is $2\times 0.0162=0.0324$.
• Conclusion (Test 1): Since $0.0324<0.05$, we reject the null hypothesis. There is a statistically significant preference for the Plunge bra over the Balconette.

Test 2: Plunge (A) vs. Wireless (C)

• Hypotheses:
  • $H_0:p=0.5$ (No preference between Plunge and Wireless)
  • $H_1:p\ne 0.5$ (A significant preference exists)
• Data: $n=16$, with $k=12$ preferences for the Plunge bra.
• p-value Calculation:
  The observed count is $k=12$. We calculate $P(X\ge 12)$ and double it. $$P(X\ge 12)=P(12)+P(13)+P(14)+P(15)+P(16)$$ $$P(X\ge 12)\approx 0.0278+0.0085+0.0018+0.0002+0.0000=0.0383$$ The two-sided p-value is $2\times 0.0383=0.0766$.
• Conclusion (Test 2): Since $0.0766>0.05$, we fail to reject the null hypothesis. Although the model preferred the Plunge bra, the preference is not strong enough to be statistically significant.

Test 3: Balconette (B) vs. Wireless (C)

• Hypotheses:
  • $H_0:p=0.5$ (No preference between Balconette and Wireless)
  • $H_1:p\ne 0.5$ (A significant preference exists)
• Data: $n=15$, with $k=11$ preferences for the Wireless bralette.
• p-value Calculation:
  The observed count is $k=11$. We calculate $P(X\ge 11)$ and double it. $$P(X\ge 11)=P(11)+P(12)+P(13)+P(14)+P(15)$$ $$P(X\ge 11)\approx 0.0417+0.0139+0.0032+0.0005+0.0000=0.0593$$ The two-sided p-value is $2\times 0.0593=0.1186$.
• Conclusion (Test 3): Since $0.1186>0.05$, we fail to reject the null hypothesis. There is no statistically significant preference between the Balconette and the Wireless bralette.

Final Synthesis and Recommendation

Let's summarize our findings:

1. Plunge > Balconette (Significant preference)
2. Plunge $\approx$ Wireless (No significant preference)
3. Balconette $\approx$ Wireless (No significant preference)

From this, we can draw a clear, practical conclusion. The Plunge bra (Style A) is the strongest candidate. It is the only style that demonstrated a statistically significant victory over another. The Balconette bra (Style B) appears to be the weakest, as it lost significantly to the Plunge and showed no advantage over the Wireless bralette.

The final ranking is Plunge > Wireless > Balconette, though only the difference between Plunge and Balconette is statistically significant at our chosen alpha level. Your final recommendation would be to feature the Plunge bra.

It's also worth noting an interesting potential outcome in these situations. It's mathematically possible to find that A > B, B > C, and C > A. This cyclical, non-transitive preference is known as the Condorcet Paradox. In such a scenario, there is no single "best" choice, and the winner depends entirely on which pair you are looking at—a concept you might recall from game theory.

Conclusion

You have now successfully navigated a multi-comparison problem using a series of sign tests. This powerful and practical technique allows you to move beyond simple A/B decisions and identify a leading candidate from a larger pool of options, a common task in style optimization.

Key Takeaways:

• To compare three or more items using sign tests, conduct a pairwise "round-robin" tournament, testing each pair individually.
• The procedure for each individual sign test remains the same: state hypotheses, count preferences, and calculate a p-value using the binomial distribution.
• The final, and most crucial, step is to synthesize the results of all tests to identify an overall winner, loser, or ranking.
• Be aware that results may not always be clear-cut; statistical significance provides a rigorous basis for your decision, but the pattern of both significant and non-significant results tells the full story.

In this module, we have focused on hypothesis testing to make definitive yes/no decisions based on collected data. However, another powerful way to use data is to iteratively update our beliefs as new information comes in. In our next module, we will explore this approach with Bayesian Inference, learning how to refine the probability that a style is a 'good fit' with each new tryout in the fitting room.