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 184.0 seconds and a standard deviation of 49.0 seconds. Use the 68-95-99.7 rule to find the indicated quantity.
To find the indicated quantity using the 68-95-99
To find the indicated quantity using the 68-95-99.7 rule, we will need to determine the range of values that fall within 1, 2, and 3 standard deviations from the mean.
1. Within 1 standard deviation:
We know that approximately 68% of the data falls within 1 standard deviation of the mean. To find the range of values within 1 standard deviation of the mean, we can use the formula: mean ± (1 * standard deviation).
The lower bound would be: 184.0 – (1 * 49.0) = 135.0 seconds.
The upper bound would be: 184.0 + (1 * 49.0) = 233.0 seconds.
Therefore, approximately 68% of individuals in the control group have eye contact times between 135.0 and 233.0 seconds.
2. Within 2 standard deviations:
Approximately 95% of the data falls within 2 standard deviations of the mean. To find the range of values within 2 standard deviations, we use the formula: mean ± (2 * standard deviation).
The lower bound would be: 184.0 – (2 * 49.0) = 86.0 seconds.
The upper bound would be: 184.0 + (2 * 49.0) = 282.0 seconds.
Therefore, approximately 95% of individuals in the control group have eye contact times between 86.0 and 282.0 seconds.
3. Within 3 standard deviations:
Approximately 99.7% of the data falls within 3 standard deviations of the mean. To find the range of values within 3 standard deviations, we use the formula: mean ± (3 * standard deviation).
The lower bound would be: 184.0 – (3 * 49.0) = 37.0 seconds.
The upper bound would be: 184.0 + (3 * 49.0) = 331.0 seconds.
Therefore, approximately 99.7% of individuals in the control group have eye contact times between 37.0 and 331.0 seconds.
By using the 68-95-99.7 rule, we can determine the range of values within specific standard deviations from the mean and estimate the proportion of data falling within those ranges.
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