Question

Time taken by a construction company to construct a flyover is a normal variate with mean 400 labour days and standard deviation of 100 labour days

Answer 1

Correct question is "Time taken by a construction company to construct a flyover is a normal variate with mean 400 labour days and a standard deviation of 100 labour days. If the company promises to construct the flyover in 450 days or less and agree to pay a penalty of Rs. 10,000 for each labour day spent in excess of 450. What is the probability that (i) the company pays a penalty of at least Rs. 2,00,000? (ii) the company takes at most 500 days to complete the flyover?"

Answer:

(i) The company pays a penalty of at least Rs. 2,00,000 is 0.242.

(ii) The company takes at most 500 days to complete the flyover 0.8413.

Step-by-step explanation:

Let X be the normal variate denoting the number of labour days. Given mean $\mu = 400$ and s.d $\sigma = 100$

(i) The company pays penalty of Rs. 2,00,000 at the rate of ₹ 10,000 per each extra labour day. No. of extra days = $\frac{200000}{10000} = 20$

So the probability that the company pays a penalty of at least Rs. 2,00,000 is probability that labour days should be at least

$450 + 20 = 470$

days, (i.e.) P ($X \leqslant 400$) Now P(X $\geqslant 470$) = P(Z ≥ (470- 400)/100) = P(Z $\geqslant 0.7$) = 0.5 – P(0 $\leqslant$Z $\leqslant$ 0.7) = 0.5 – 0.258 = 0.242

(ii)

$P (X \leqslant 500) \\ = P(Z \leqslant \frac{500 - 40}{100} ) \\ = P (Z \leqslant 1) P(Z \leqslant 1) \\ = 0.5 + P (0 \leqslant Z \leqslant 1) \\ = 0.5 + 0.3413 \\ = 0.8413$

Thus the probability that the company takes at most 500 days to complete the flyover is 0.8413.

This is a Mathematics problem of standard deviation related.

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