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Question 202:



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a) First, find the z-score for 280 days by subtracting it from the mean and dividing by the standard deviation (280-250)/25 = 1.2. The one-sided area associated with a z-score of 1.2 can be looked up in the z-score to percentile calculator to get ~88.5%. Since the question wants to know the probability the site will not be completed, its 100% minus the probability that it will be completed, which we calculated as 88.5, so there is roughly a 11.5% chance the site will not be completed within 280 days.

b) Use the same approach as in a except this question is asking for two-sided area since it asks for the probability above and below two values (which are both 25 days above the mean). So we can take one value and calculate the z (275-250)/25= 1, and the two sided area is ~68% so the area above and below this level tells us there is around a 32% probability the site will take either longer than 275 days or less than 225 days.

c)We need to find the z-score associated with 99% of the area under the normal curve. So to find that, we look up .99 in the percentile to z-score calculator or use the z-table in a back of a statistics text. You should get a z-score of approximately 2.33. So now we setup the equation to find the number of days that are 2.33 standard deviations (z-scores) above the mean:

  1. (x-250)/25 = 2.33
  2. x-250 = 58.25
  3. x = 308.25

So it would take 308.25 workdays before 99% of the houses would be complete.

d). We need to find the area between two z-scores. To do this we calculate the area for the larger and subtract out the area of the smaller z-score.  We already calculated the z-score for 280 days in part a and got a z-score of 1.2 which represents 88.5 percent of the area. For 260 days the z-score is (260-250)/25 = .4. The area associated with a z-score of .4 from the z-score to percentile calculator is 65.54. So the area in between is 88.5- 65.54 = 22.96, or there is about a 23% chance the houses will be completed between 260 and 280 days.

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