Question

A company has a customer services call centre. The company believes that the time taken to complete a call to the call centre may be modelled by a normal distribution with mean 16 minutes and standard deviation σ minutes.

 

Given that 10% of the calls take longer than 22 minutes,

 

(a)   show that, to 3 significant figures, the value of σ is 4.68.

(3)

 

(b)  Calculate the percentage of calls that take less than 13 minutes.

(1)

 

A supervisor in the call centre claims that the mean call time is less than 16 minutes. He collects data on his own call times.

 

·         20% of the supervisor’s calls take more than 17 minutes to complete.

 

·         10% of the supervisor’s calls take less than 8 minutes to complete.

 

Assuming that the time the supervisor takes to complete a call may be modelled by a normal distribution,

 

(c)  estimate the mean and the standard deviation of the time taken by the supervisor to complete a call.

(6)

 

(d)   State, giving a reason, whether or not the calculations in part (c) support the supervisor’s claim.

(1)

(Total for Question 4 is 11 marks)

Answer 1

(a) show that, to 3 significant figures, the value of σ is 4.68

[P(X > 22) > 0.1]

[P(X > 22) > 0.1]=Z_0.10

[P(X > 22) > 0.1]=1.28

(22-16)/σ = 1.28

(22-16)/1.28 = σ

4.6875 =σ

b)

P(X < 13) = 0.2611=26.1%

c)

P(X > 17) = 0.2 or P(X < 8) = 0.1

X-μ/σ =Z_0.2 or  X-μ/σ =Z_0.1

X-μ/σ =0.8416 or  X-μ/σ = −1.2816

17 −μ = 0.8416σ or  −(8 − μ = −1.2816σ)

σ  = 4.238

μ= 13.432

d)

μ = 13.4 < 16

Yes, supports supervisor’s belief