2.21 The financial services call center in Problem 2.20 also...

2.21 The financial services call center in Problem 2.20 also monitors call duration, the amount of time spent speaking to customers on the phone. The file Call Duration contains the following data for time, in seconds, spent by agents talking to 50 customers: 243 290 199 240 125 151 158 66 350 1141 251 385 239 139 181 111 136 250 313 154 78 264 123 314 135 99 420 112 239 208 65 133 213 229 154 377 69 170 261 230 273 288 180 296 235 243 167 227 384 331 a. Construct a frequency distribution and a percentage distribution. b. Construct a cumulative percentage distribution. c. What can you conclude about call center performance if a call duration target of less than 240 seconds is set?

Transcript

00:01 In this question, we are going to use the exponential distribution, which is defined as the probability distribution that describes the time between events which occur continuously and independently at an average time.

00:15 The probability density function for this distribution is f of x equal lambda e power negative lambda x or 0.

00:29 For lambda e power negative x, lambda x, its range from x bigger than or equal 0, and 0 will be 0 elsewhere.

00:47 Let.

00:48 X, let x has exponential distribution with parameter lambda.

00:55 Then ex will equal to 1 over lambda and varx will equal to 1 over lambda square.

01:03 We have one call every two minutes, so we can say that average time for one call would equal to two minutes.

01:12 The time between calls is exponential distributed, so we can say that let x be time between two calls.

01:30 X also has exponential distribution with average 2 x has also exponential distribution with average time 2 and we need to find the parameter lambda so we can get it as follows 2 equal x equal 1 over lambda from this we can say that lambda would equal 2 half 1 over 2 so x approximately be exponential half now we need to find the average time for 5 calls we need average time for 5 calls we know that for 1 minute when we finish it for a second we also need on average two minutes so for two calls we need time what equal to two by two equal four minutes now for five calls we need five by two what equal to 10 minutes.

03:40 For the other question, b, we have to find the probability density function.

04:01 For x, f of x, equal to half by e.

04:15 Power negative half x, for x bigger than equal zero, and zero elsewhere.

04:25 Now we need to find the probability that we need more three minutes to take the next call.

04:43 So, the probability for more three minutes to take the next call will be probability of x.

05:32 For x bigger than 3 will equal integration from 3 to infinity f of x d x will equal to integration from 3 to infinity for equation half e power negative half x d t d t so by substitution we can see that u equal to half x so d u will equal to half it will be equal integration from one and half to infinity e power negative u d u d u d u after integration we'll get negative e power negative u from 1 .5 to infinity will equal to negative.

06:53 0 minus e power negative 1 .5 will equal to 0 .23.

07:10 So we can say that the probability that next call will come more than 3 minutes is 0 .223...

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