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25 Sep 2025, 03:53
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At least half of respondents gave at least 4 stars for at least two features.: Yes
At least eight respondents gave Search 1 star and Support 3 stars.: Yes
In exactly two features, the percentage at 4 stars exceeds the percentage at 2 stars.: No
The table shows the results of a poll in which more than 100 respondents rated five features of a video-streaming app on a 1-to-5 star scale (1 being the lowest and 5 the highest). Each respondent gave one rating for each feature.
For each statement, select Yes if it is true based on the table. Otherwise, select No.
| Rating | Home | Search | Playback | Notifications | Support |
|---|---|---|---|---|---|
| 5 stars | 24% | 3% | 53% | 18% | 4% |
| 4 stars | 35% | 8% | 27% | 31% | 10% |
| 3 stars | 22% | 15% | 10% | 23% | 56% |
| 2 stars | 12% | 22% | 6% | 15% | 20% |
| 1 star | 7% | 52% | 4% | 13% | 10% |
For each statement, select Yes if it is true based on the table. Otherwise, select No.
| Yes | No | |
| At least half of respondents gave at least 4 stars for at least two features. | ||
| At least eight respondents gave Search 1 star and Support 3 stars. | ||
| In exactly two features, the percentage at 4 stars exceeds the percentage at 2 stars. |
ShowHide Answer
Official Answer
At least half of respondents gave at least 4 stars for at least two features.: Yes
At least eight respondents gave Search 1 star and Support 3 stars.: Yes
In exactly two features, the percentage at 4 stars exceeds the percentage at 2 stars.: No
25 Sep 2025, 03:53
Kudos
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Official Solution:
• At least half of respondents gave at least 4 stars for at least two features.
“At least 4 stars” means combining the percentages of 4-star and 5-star ratings. We need to check whether two or more features have this combined percentage above 50. Playback clearly does (53% + 27% = 80%), and Home also does (24% + 35% = 59%). That makes two features, so the condition is satisfied.
Answer: Yes
• At least eight respondents gave Search 1 star and Support 3 stars.
Search has 52% at 1 star and Support has 56% at 3 stars. When the same group of people rates both features, the minimum guaranteed overlap of these two groups is 52 + 56 − 100 = 8%. That means at least 8% of respondents must have chosen both Search 1 star and Support 3 stars. Since the poll included more than 100 people, 8% corresponds to at least 8 individuals. Therefore, the statement is correct.
Answer: Yes
• In exactly two features, the percentage at 4 stars exceeds the percentage at 2 stars.
Comparing 4 vs 2 stars: Home 35 vs 12 (yes), Search 8 vs 22 (no), Playback 27 vs 6 (yes), Notifications 31 vs 15 (yes), Support 10 vs 20 (no). This happens in three features, not exactly two.
Answer: No
Correct answer:
At least half of respondents gave at least 4 stars for at least two features. "Yes"
At least eight respondents gave Search 1 star and Support 3 stars. "Yes"
In exactly two features, the percentage at 4 stars exceeds the percentage at 2 stars. "No"
bb
• At least half of respondents gave at least 4 stars for at least two features.
“At least 4 stars” means combining the percentages of 4-star and 5-star ratings. We need to check whether two or more features have this combined percentage above 50. Playback clearly does (53% + 27% = 80%), and Home also does (24% + 35% = 59%). That makes two features, so the condition is satisfied.
Answer: Yes
• At least eight respondents gave Search 1 star and Support 3 stars.
Search has 52% at 1 star and Support has 56% at 3 stars. When the same group of people rates both features, the minimum guaranteed overlap of these two groups is 52 + 56 − 100 = 8%. That means at least 8% of respondents must have chosen both Search 1 star and Support 3 stars. Since the poll included more than 100 people, 8% corresponds to at least 8 individuals. Therefore, the statement is correct.
Answer: Yes
• In exactly two features, the percentage at 4 stars exceeds the percentage at 2 stars.
Comparing 4 vs 2 stars: Home 35 vs 12 (yes), Search 8 vs 22 (no), Playback 27 vs 6 (yes), Notifications 31 vs 15 (yes), Support 10 vs 20 (no). This happens in three features, not exactly two.
Answer: No
Correct answer:
At least half of respondents gave at least 4 stars for at least two features. "Yes"
At least eight respondents gave Search 1 star and Support 3 stars. "Yes"
In exactly two features, the percentage at 4 stars exceeds the percentage at 2 stars. "No"
15 Nov 2025, 04:44
Kudos
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This is a great question that’s helpful for learning and I don’t quite agree with the solution. Statement 1: At least half of respondents gave at least 4 stars for at least two features.
No.
This requires the proportion of respondents rating ≥4 stars on ≥2 features to be ≥50%. The table gives per-feature proportions for ≥4 stars:
Total "high ratings" across all: 59% + 11% + 80% + 49% + 14% =
Let r = proportion with ≥2 highs.
The remaining (1 - r) can cover at most (1 - r) highs (one each via singletons; zeros cover none).
Thus, the r must cover at least 213% - (1 - r)% = 112% + r% of the highs.
Each of the r can cover up to 5 highs, but capped by per-feature proportions: max coverage by r is ∑ min(r, pi) where pi are the ≥4% per feature.
We solve for minimal r where ∑ min(r, pi) ≥ 1.13 + r.
For r in [0.14, 0.49] (relevant range): ∑ min(r, pi) = 0.11 + 0.14 + 3r = 0.25 + 3r.
Set 0.25 + 3r = 1.13 + r → 2r = 0.88 → r = 0.44.
This satisfies the caps (0.44 < 0.49, 0.59, 0.80).
Thus, it's possible for only 44% to have ≥2 highs
No.
This requires the proportion of respondents rating ≥4 stars on ≥2 features to be ≥50%. The table gives per-feature proportions for ≥4 stars:
Total "high ratings" across all: 59% + 11% + 80% + 49% + 14% =
Let r = proportion with ≥2 highs.
The remaining (1 - r) can cover at most (1 - r) highs (one each via singletons; zeros cover none).
Thus, the r must cover at least 213% - (1 - r)% = 112% + r% of the highs.
Each of the r can cover up to 5 highs, but capped by per-feature proportions: max coverage by r is ∑ min(r, pi) where pi are the ≥4% per feature.
We solve for minimal r where ∑ min(r, pi) ≥ 1.13 + r.
For r in [0.14, 0.49] (relevant range): ∑ min(r, pi) = 0.11 + 0.14 + 3r = 0.25 + 3r.
Set 0.25 + 3r = 1.13 + r → 2r = 0.88 → r = 0.44.
This satisfies the caps (0.44 < 0.49, 0.59, 0.80).
Thus, it's possible for only 44% to have ≥2 highs
