Hi Blake,
To answer the three questions, just follow the simple 2-step procedure. First, calculate your z-score. For the first question, plug the numbers into this formula: Z = (your score - mean score) / SD.
You want to know the probability that the randomly selected person's weight is 71 or heavier. Therefore,
Z = (71 - 68) / 4 = 0.75.
Look up the proportion corresponding to this Z-score in your normal distribution table at the back of your textbook. Any statistics textbook has this in the appendix. The area beyond the z-score is 0.2266. In probabilistic terms, multiply it by 100, and the probability of a person weighing over 71 kg is 22.66%.
Basically, apply the same logic for your second and third questions. For the second question, first calculate the area between 67 and the mean (which is 68). The z-score for that area is -0.25. The proportion corresponding to the z-score of -0.25 is 0.09. And calculate the area between the mean (68) and 83. The z-score is 3.75. The area corresponding to a z-score of 3.75 is 0.4999.
Just add them up: 0.09 + 0.4999. Then the total area is 0.5986. In probabilistic terms, the probability that the randomly selected person weighs between 67kg and 83kg is 59.86%.
And to calculate the probability that the randomly selected person weighs less than 75kg, calculate the z-score, which is (75-68) / 4 = 7/4. So, the z-score is 1.75. Look up the area corresponding to a z-score of 1.75 in the normal distribution table. You will see that the area between the mean (68) and 75 is 0.4599. You already know that half the total area is 50%. So, 50% + 45.99% = 95.99%. The probability that the randomly selected person will weigh less than 75kg is 95.99%.
