Question

In a study of facial​ behavior, people in a control group are timed for eye contact in a​ 5-minute period. Their times are normally distributed with a mean of 56.0 seconds and a standard deviation of 181.0 seconds. Use the​ 68-95-99.7 rule to find the indicated quantity. Find the percentage of times within 56.0 seconds of the mean of 181. 0 seconds.

Answer 1

To find the percentage of times within 1 standard deviation of the mean, we can use the 68-95-99.7 rule, also known as the empirical rule. According to this rule:

- Approximately 68% of the data falls within 1 standard deviation of the mean.
- Approximately 95% of the data falls within 2 standard deviations of the mean.
- Approximately 99.7% of the data falls within 3 standard deviations of the mean.

In this case, we want to find the percentage of times within 1 standard deviation of the mean (56.0 seconds), which means we need to calculate the range from the mean minus 1 standard deviation to the mean plus 1 standard deviation.

Lower limit = Mean - 1 * Standard deviation = 56.0 - 181.0 = -125.0 seconds
Upper limit = Mean + 1 * Standard deviation = 56.0 + 181.0 = 237.0 seconds

Now we need to find the percentage of times within this range. Since the data is normally distributed, we know that approximately 68% falls within this range according to the 68-95-99.7 rule.

Therefore, the percentage of times within 56.0 seconds of the mean of 181.0 seconds is approximately 68%.