Question 5 OR (d)
Percentage of customers rating a restaurant above 3.3 stars
5 marks
- Time to write
- About 10 minutes
- Topic
- Normal distribution
Question
Question 5 OR (d) · 5 marksThe online reviews received by a restaurant have ratings that follow a normal distribution with an average score of 3.8 stars (out of 5), with a standard deviation of 0.5 stars. What percentage of the customers give a rating higher than 3.3 stars?
[Given: Table values of the area: Z = 0.5 gives 0.1915 and Z = 1 gives 0.3413]
Reworded from the university paper: names and figures differ, the task is the same. The original paper belongs to the University of Delhi. See DU's question bank.
Before you write
Read off the question and its marks
Write it in this order
- Read the data; note exactly what is asked 1 min
- 1 Standardisation 1½ marks 2 min
- 2 Area from the table, using symmetry 2½ marks 4 min
- 3 Percentage of customers 1 mark 2 min
- Check: arithmetic, units (₹, %), final answer underlined 1 min
Short on time? Step 2 carries 2½ of the 5 marks: guard its minutes.
Try it first. About 10 minutes for 5 marks. Write it out, then open the solution.
Solution
Hover or tap a marked figure, like 72,000, to see how it is made and where else it appears.
Standardising. Ratings are normal with μ = 3.8 stars and σ = 0.5 stars. For X = 3.3, z = (X − μ) / σ = (3.3 − 3.8) / 0.5 = −1, using the standard normal variate.
Area required. A rating higher than 3.3 means z > −1. This covers the area from z = −1 up to the mean plus the whole right half of the curve. By symmetry, the area from −1 to 0 equals the tabled area from 0 to 1, 0.3413, and each half of the curve has area 0.5.
| Part of the curve | Area |
|---|---|
| z = 0 and above (right half) | 0.5 |
| z = −1 to 0 | 0.3413 |
| z above −1 | 0.8413 |
Percentage of customers. P(X > 3.3) = 0.5 + 0.3413 = 0.8413, so 84.13% of customers give a rating higher than 3.3 stars. (The value given for z = 0.5 is not needed.)
Under the normal curve: the shaded area starts at z = −1, one SD to the left of the mean, and runs to the right end of the curve.
Where the 5 marks sit
| # | Component | Marks |
|---|---|---|
| 1 | Standardisation | 1.5 |
| 2 | Area from the table, using symmetry | 2.5 |
| 3 | Percentage of customers | 1 |
| Total | 5 |
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