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Bulb life of two companies: grouped data

Short answer

Means: A = 1095.7 h, B = 1134.1 h. Sample std dev: A = 376.1 h, B = 362.2 h. Coefficients of variation: A = 34.3%, B = 31.9%, so company B’s bulbs are more uniform.

The samples of length of life of bulbs from two companies are given below.

Length of life (hours)Company ACompany B
500-600103
600-700218
700-800615
800-900812
900-1000214
1000-1100105
1100-1200215
1200-13001213
1300-1400197
1400-150097
1500-160034
1600-170076
1700-180053
1800-190042
1900-200013
a) Calculate mean length of life of bulbs for company A and company B.
b) Calculate sample standard deviation and sample variance for given data.
c) Which company's bulbs are more uniform?

Use class mid-points m for the grouped data. Mean = Σfm/N; sample variance s² = Σf(m − x̄)²/(N − 1) = [Σfm² − N·x̄²]/(N − 1).

Classmf (A)f (B)
500–600550103
600–700650218
700–800750615
800–900850812
900–1000950214
1000–11001050105
1100–12001150215
1200–130012501213
1300–14001350197
1400–1500145097
1500–1600155034
1600–1700165076
1700–1800175053
1800–1900185042
1900–2000195013
Total138107

Σfm: A = 151200, B = 121350. Σfm²: A = 185045000, B = 151527500.

a) Mean life

  • Company A: x̄ = 151200/138 = 1095.65 hours
  • Company B: x̄ = 121350/107 = 1134.11 hours

b) Sample variance and standard deviation

  • Company A: s² = (185045000 − 138×1095.65²)/137 = 141477.3, s = 376.13 hours
  • Company B: s² = (151527500 − 107×1134.11²)/106 = 131160.3, s = 362.16 hours

c) Uniformity

Coefficient of variation CV = (s/x̄) × 100:

  • A: 376.13/1095.65 × 100 = 34.33%
  • B: 362.16/1134.11 × 100 = 31.93%

Company B's bulbs are more uniform (smaller CV, less relative variation in life).

Answer: x̄_A = 1095.7 h, x̄_B = 1134.1 h; s_A = 376.1 h, s_B = 362.2 h; B is more uniform.

✓ Answer checked by Claude Sonnet 5.5

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