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Question 517:



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We will want to calculate z-scores to answer these questions. A z-score is the number of standard deviations above or below the mean using the standard normal curve. We need to convert this normal distribution with a mean of 35 minutes and sd of 8 minutes to standard normal form by subtracting the mean from the data-point and dividing that result by the standard deviation.

  1. (30-35)/8 = a z-score of -.625. Now we find the area under the standard normal curve that is .625 standard deviations above the mean. To do this we use a z-table or the z-score to percentile calculator and select 1-sided area since we want to know area below 30 minutes. This gets us a percent of 26.6, or 26.6% of days will take less than 30 minutes
  2. This is a between area question and we need to find the area of the larger area and subtract the smaller area for the times of 50 and 40 minutes. The large area is (50-35)/8 = a z-score of 1.875 and the smaller area is  (40-35)/8 = .625. The larger z-score has a 1-sided area of  96.96% and the smaller area is 73.4%. Subtracting the smaller we get an area of 96.96-73.4 = 23.56%. So on 23.6% of days it will take between 40 and 50 minutes.
  3. This one is just like part 2, except we're interested in between 30 and 45 minutes. For z-scores we get (30-35)/8 = -.625 and (45-35)/8 = 1.25. The 1-sided areas are 26.6 and 89.4%. Subtracting the two we get 89.4-26.6 = 62.8%, or on 62.8% of days it will take between 30 and 45 minutes.
  4. The longest 10% is the same as finding the area above 90% for 1-sided area. We just need to find the z-score first associated with 90%. To do so, we can use the percentile to z-score calculator and select 1-sided area. We get a z-score of 1.28. Now we setup a simple equation to solve for the unknown time that is 1.2817 standard deviations above the mean.
    1. (x-35)/8 = 1.2817
    2. x-35 = 10.2536
    3. x = 45.2536
    4. So, the longest 10% of trips will take above 45.25 minutes to get to work.

Finally, you can visualize the area involved by using the interactive graph of the standard normal curve enter the mean of 35 and standard deviation of 8 in the second graph, then hover over the area and z-scores for the questions above.

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