I29-146

Question

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.
      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 .

Bunuel · Answer

Official Solution:

• At least two features were rated 4 stars or higher by at least half of the respondents. 
 
“At least 4 stars” means combining the percentages of 4-star and 5-star ratings. We need to check whether at least two features were rated 4 stars or higher by at least half of the respondents. Playback clearly was (53% + 27% = 80%), and Home also was (24% + 35% = 59%). That gives us at least 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 two features were rated 4 stars or higher by at least half of the respondents.  "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"

NeerajR2000 · Answer

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   (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

Bunuel · Answer

I’m not fully following what you’re trying to say there, but Statement 1 is much simpler than what you’re turning it into.

It does  not  ask how many people gave two high ratings.
 It only asks whether there are  two features  where at least half of respondents gave 4 or 5 stars.

Home and Playback already cross 50 percent. That alone makes the statement true.

Karym · Answer

I like the solution - it’s helpful.

Mahi2804 · Answer

I like the solution - it’s helpful.

Karym · Answer

I like the solution - it’s helpful.

ayanb21 · Answer

I like the solution - it’s helpful.

Aaryan911 · Answer

Bunuel  
I think the st1 is incorrectly framed in that case, i posted the same statement on chatgpt with the following prompt -: 
 "Explain the meaning of this statment - At least half of respondents gave at least 4 stars for at least two features." 
 Heres the response -  
  There are three parts:  
 
    At least half of respondents

If there are 100 respondents,  50 or more  satisfy the condition.    
 
    Gave at least 4 stars

A rating of  4 or 5 stars  (assuming a 5-star scale).    
 
    For at least two features

Each of those respondents rated  two or more features  with 4 or 5 stars.

What it does  NOT  mean
A common mistake is to think it means:

❌ At least half of respondents gave  every feature  at least 4 stars.  
  ❌ At least two features received at least 4 stars from half the respondents.

Bunuel · Answer

Thank you for pointing this out. I edited the statement and adjusted the solution to remove the ambiguity.

I29-146

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My Rewards

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%