Distribution of the amount of monthly utility bill of a 3-bedroom house using gas or electric energy had μ = $150 and σ = $10. What percentage of homes will have a monthly utility bill between $130 and $170? What percentage of homes will have a monthly utility bill of less than $120 or more than $180?

Gemini 3.5 Flash

Answer: Assuming a normal distribution, approximately 95% of homes will have a monthly utility bill between $130 and $170, and approximately 0.3% will have a bill of less than $120 or more than $180.

All models in this roundtable are in complete agreement, which confirms the mathematical correctness of the solution based on the Empirical Rule (68-95-99.7 rule) for normal distributions. First, for the range between $130 and $170, these values correspond exactly to two standard deviations below and above the mean (μ ± 2σ), where μ = $150 and σ = $10. According to the Empirical Rule, approximately 95% of the distribution falls within this range. Second, for bills less than $120 or more than $180, these boundaries represent three standard deviations from the mean (μ ± 3σ). Since approximately 99.7% of homes fall within three standard deviations, the remaining percentage in the outer tails is 100% - 99.7% = 0.3% (or more precisely, 0.27% using exact z-tables).

Grok 4.3

Answer: Approximately 95% of homes will have monthly utility bills between $130 and $170, while about 0.3% will have bills less than $120 or more than $180.

All other models in this round (GPT-5.5, Gemini 3.5 Flash, and Claude Opus 4.7) reached the identical conclusion using the empirical rule for a normal distribution. The interval $130 to $170 is exactly μ ± 2σ, so the 68-95-99.7 rule directly gives the 95% figure; likewise, $120 and $180 are μ ± 3σ, yielding the complementary 0.3% outside that range. Because the other responses confirm the same application of the empirical rule without introducing contradictory evidence or alternative assumptions, my original position remains unchanged.

GPT-5.5

Answer: Assuming the utility bills are normally distributed, about 95% of homes will have bills between $130 and $170, and about 0.3% will have bills less than $120 or more than $180.

I maintain my previous position because the other models give the same standard solution and do not introduce a reason to change it. The mean is μ = $150 and the standard deviation is σ = $10. A bill of $130 is 2 standard deviations below the mean, since ($130 − $150) / $10 = −2, and a bill of $170 is 2 standard deviations above the mean, since ($170 − $150) / $10 = 2. By the Empirical Rule for a normal distribution, approximately 95% of observations fall within 2 standard deviations of the mean. More precisely, this is about 95.45%. For the second part, $120 is 3 standard deviations below the mean and $180 is 3 standard deviations above the mean. The Empirical Rule says about 99.7% of observations fall within 3 standard deviations, so the percentage outside that range is about 100% − 99.7% = 0.3%. More precisely, using normal-table values, it is about 0.27% total in both tails. So the expected answers are approximately 95% between $130 and $170, and approximately 0.3% below $120 or above $180, assuming a normal distribution.

Claude Opus 4.7

Answer: Approximately 95% of homes will have a monthly utility bill between $130 and $170, and approximately 0.3% of homes will have a bill of less than $120 or more than $180.

All four models, including myself, arrived at the same answer using the Empirical Rule (68-95-99.7 Rule) for a normal distribution. **Part 1: Between $130 and $170** - $130 = μ − 2σ (z = −2) - $170 = μ + 2σ (z = +2) The Empirical Rule states that approximately 95% of values in a normal distribution lie within 2 standard deviations of the mean, so about 95% of homes fall in this range. **Part 2: Less than $120 or more than $180** - $120 = μ − 3σ (z = −3) - $180 = μ + 3σ (z = +3) Approximately 99.7% of values lie within 3 standard deviations of the mean, leaving about 100% − 99.7% = 0.3% in the combined tails (roughly 0.15% on each side). The other models (Grok 4.3, GPT-5.5, and Gemini 3.5 Flash) reached identical conclusions. Some noted the more precise normal-table values of 95.45% and 0.27%, which I agree are technically more accurate, but the Empirical Rule values of 95% and 0.3% are the standard expected answers for this type of problem. I am maintaining my position because the consensus confirms the methodology and answer.

Final Summary

The AI Roundtable reached a perfect consensus by applying the 68-95-99.7 Empirical Rule for normal distributions. All four models, including GPT-5.5 and Gemini 3.5 Flash, identified that the specified price ranges correspond exactly to two and three standard deviations from the mean. The group concluded that the statistical likelihood of extreme utility bills is remarkably low under these parameters.

All 4 models agreed