Worked solution
Estimate a defect probability from a frequency table
A company manufactures a wide variety of components for use in domestic appliances.
The company introduces a new type of light fitting for refrigerators. Each day a supervisor selects a sample of 100 of the light fittings for testing.
(i) How should the light fittings be selected? Give a reason for this method of selection.
The supervisor records the number of faulty light fittings found on each of 150 working days. Her results are shown in the table.
| Number faulty | Number of days |
|---|---|
| 0 | 4 |
| 1 | 19 |
| 2 | 38 |
| 3 | 41 |
| 4 | 22 |
| 5 | 16 |
| 6 | 6 |
| 7 | 4 |
| 8 or more | 0 |
(ii) Use the information in the table to estimate , the probability that a light fitting is faulty.
(iii) Assuming the number of faulty fittings found each day follows the binomial distribution , find the expected number of days on which 3 faulty fittings are found in a period of 150 working days.
[2021/II/11(a)]
CJC · Discussion · Question 10
Answer
(i) Random selection with equal chances, to avoid selection bias. (ii) . (iii) days.
(i)
Take a random sample, giving each fitting an equal chance of selection, to avoid systematic selection bias. The source also specifies independent selections; independence/common defect probability are modelling assumptions supporting the later binomial calculation. For a finite sample selected without replacement, equal selection chances alone do not imply exact independence.
(ii)
The total faults recorded are . The frequencies total 150 days; the 8-or-more category has frequency zero, so no missing tail value is needed. Mean faults per day =450/150=3. Since np=3 with n=100, the estimate is p=0.03.
(iii)
Let count faulty fittings in a daily sample. Then . The expected number of days is 150 times this probability: 34.12111912…=34.1 days (3 s.f.). An expected frequency need not be an integer.
Common mistakes
The table counts days, not fittings. The zero-frequency “8 or more” category must be retained.
Solution notes
The source table is transposed in the student presentation for narrow screens; every category and frequency is unchanged. The original two-row table is retained in the native record.
Source sampling wording includes independent selection. The reviewed answer distinguishes that model premise from equal-chance finite sampling without replacement.