## Question 385:

1

Calculate z-scores to answer this question by subtracting the data-point from the mean and divide that result by the standard deviation. From the z-score you can then look-up the proportion of area under the normal curve to answer the questions using the z-score to percentile calculator. Depending on what the question is asking you will either use the 1-sided or 2-sided area.

1. <= 12km : (12-10)/3 = a z-score of .666. Since this is asking within 12km we want the area up to .666 or the 1-sided area. Since the z-score is positive we will want the larger area = 74.7% of the area.
2. > 15km : (15-10)/3 = 1.666 and we want the smaller area since it is asking for the area NOT under the curve = 4.8%
3. The closest 10% have a 1-sided z-score of -1.28. Setting up an equation:
1. (x-10)/3 = -1.28
2. x-10 = 3*-1.28
3. x = 10-3.84
4. x = 6.16
5. So the closest 10% live within 6.16km.
4. Finally, the furthest 33% (or the closest 66%) have a 1-sided z-score of .4124. Setting up an equation:
1. (x-10)/3 = .412
2. x-10 = 3*.412
3. x = 10+1.236
4. x = 11.236
5. So 33% live further than 11.236 km away.

You can visualize this relationship using the interactive graph of the standard normal curve. Enter the mean of 10 and standard deviation of 3 in the graphs to see the areas.

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