Final answer:
To draw a normal curve and label standard deviations, use the empirical rule. Percentages of values within and outside specific intervals can be calculated using the rule as well.
Step-by-step explanation:
(a) To draw a normal curve and label 1, 2, and 3 standard deviations on both sides of the mean, you can use the empirical rule. The empirical rule states that approximately 68% of values fall within 1 standard deviation, 95% fall within 2 standard deviations, and 99.7% fall within 3 standard deviations of the mean. So, for a normal distribution with a mean of 175 and a standard deviation of 37, you can calculate the values as follows:
1 standard deviation: Mean ± (1 * Standard Deviation) = 175 ± (1 * 37) = 138, 212
2 standard deviations: Mean ± (2 * Standard Deviation) = 175 ± (2 * 37) = 101, 249
3 standard deviations: Mean ± (3 * Standard Deviation) = 175 ± (3 * 37) = 64, 286
(b) To find the percentage of values within the interval (138, 212), you can use the empirical rule. Since this interval represents 1 standard deviation from the mean, approximately 68% of values fall within this range.
(c) To find the percentage of values within the interval (101, 249), you can again use the empirical rule. This interval represents 2 standard deviations from the mean, so approximately 95% of values fall within this range.
(d) Similarly, to find the percentage of values within the interval (64, 286), you can use the empirical rule. This interval represents 3 standard deviations from the mean, so approximately 99.7% of values fall within this range.
(e) To find the percentage of values outside the interval (138, 212), you can subtract the percentage within the interval from 100%. From part (b), we know that approximately 68% of values fall within 1 standard deviation of the mean, so approximately 32% of values fall outside this range.
(f) For the interval (101, 249), approximately 95% of values fall within 2 standard deviations of the mean. So, the percentage of values outside this interval is approximately 100% - 95% = 5%.
(g) Similarly, for the interval (64, 286), approximately 99.7% of values fall within 3 standard deviations of the mean. So, the percentage of values outside this interval is approximately 100% - 99.7% = 0.3%.