Great Insurance company has collected the data regarding the duration (how long it takes) to settle insurance claims. The data collected indicate that the duration follows a normal distribution with mean 28 days and standard deviation 8 days. 1. What proportion of the duration is between 20 and 40 days old? 2. What proportion of the duration is less than 30 days old? 3. What is the number of days in which 75% of all claims are above?

Question

Great Insurance company has collected the data regarding the duration (how long it takes) to settle insurance claims. The data collected indicate that the duration follows a normal distribution with mean 28 days and standard deviation 8 days.   1. What proportion of the duration is between 20 and 40 days old?   2. What proportion of the duration is less than 30 days old?   3. What is the number of days in which 75% of all claims are above?

hotcoal3z · Accepted Answer

Step 1 the PDF of normal distribution is =1/σ⋅2π⋅e−x−u2/2σ2 standard normal distribution is a normal distribution with, mean of 0, standard deviation of 1 equation of the normal curve is (Z)=x−usd∼N(0,1) mean(u)=28 standard deviation (sd)=8 Step 2 (1) to find P(a≤Z≤b)=F(b)−F(a) P(X&amp;lt;20)=20−288 =−88=−1 =P(Z&amp;lt;−1) from standard normal table =0.1587 P(X&amp;lt;40)=40−288 =128=1.5 =P(Z&amp;lt;1.5) from standard normal table =0.9332 P(20&amp;lt;X&amp;lt;40)=0.9332−0.1587=0.7745 proportion of the duration is between 20 and 40 days old is 77.45% Step 3 (2) P(X&amp;lt;30)=30−288 =28=0.25 =P(Z&amp;lt;0.25) from standard normal table =0.5987 proportion of the duration is less than 30 days old is 59.87% (3) p(z&amp;gt;x)=0.75 value of z to the cumulative probability of 0.75 from normal table is -0.6745 p(x−us.d&amp;gt;x−288)=0.75 that is, (x−288)=−0.6745 ⇒x=−0.6745⋅8+28=22.6041 the number of days in which 75% of all claims are above is at 22.6∼23days

Great Insurance company has collected the data regarding the duration

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