Answer :
a) The probability of a lag time less than 13 hours is approximately 16%.
b) The probability of a lag time between 15 and 19 hours is approximately 34%.
To answer these questions using the empirical rule, which applies to normally distributed data, we need to understand the proportions of data within certain numbers of standard deviations from the mean. The empirical rule states that:
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.
Given:
Mean (μ) = 15 hours
Standard Deviation (σ) = 2 hours
(a) Probability that lag time is less than 13 hours
To find the probability that a lag time is less than 13 hours, we calculate how many standard deviations 13 hours is from the mean.
Calculate the z-score for 13 hours:
z = (X - μ) / σ
z = (13 - 15) / 2
z = -2 / 2
z = -1
Using the empirical rule:
A z-score of -1 corresponds to 1 standard deviation below the mean.
Approximately 68% of the data falls within 1 standard deviation of the mean, so 34% falls between the mean and 1 standard deviation below the mean.
Since the normal distribution is symmetrical:
Half of the distribution is below the mean (50%).
34% is between the mean and 1 standard deviation below the mean.
Thus, the probability of being more than 1 standard deviation below the mean (lag time less than 13 hours):
50% (below mean) - 34% (between mean and 1 standard deviation below) = 16%
So, the probability is approximately 16%.
(b) Probability that lag time is between 15 and 19 hours
To find the probability that a lag time is between 15 and 19 hours:
Calculate the z-score for 19 hours:
z = (X - μ) / σ
z = (19 - 15) / 2
z = 4 / 2
z = 2
Using the empirical rule:
A z-score of 2 corresponds to 2 standard deviations above the mean.
Approximately 95% of the data falls within 2 standard deviations of the mean, meaning 47.5% falls between the mean and 2 standard deviations above the mean.
The probability of lag time less than 13 hours is 16%, and the probability of lag time between 15 and 19 hours is 47.5%, using the empirical rule and standard deviations.
(a) Probability of lag time less than 13 hours
To find the probability of a lag time less than 13 hours, we use the empirical rule (68-95-99.7 rule):
- The mean (
) is 15 hours. - The standard deviation (
) is 2 hours.
Thirteen hours is one standard deviation below the mean (15 - 2 = 13). According to the empirical rule, approximately 68% of the data falls within one standard deviation of the mean (13 to 17 hours). Since the normal distribution is symmetrical, about 34% of the data falls between the mean and one standard deviation below the mean. Thus, the probability of a lag time less than 13 hours is:
P(X < 13) = 0.5 - 0.34 = 0.16, or 16%
(b) Probability of lag time between 15 and 19 hours
To find the probability of a lag time between 15 and 19 hours, we note that 19 hours is two standard deviations above the mean (15 + 2 * 2 = 19). According to the empirical rule, approximately 95% of the data falls within two standard deviations of the mean. Thus, the probability of a lag time between 15 and 19 hours is half of this range:
P(15 < X < 19) = 0.95 / 2 = 0.475, or 47.5%
In summary, the probability that lag time is less than 13 hours is 16%, and the probability that lag time is between 15 and 19 hours is 47.5%.