Answer :
Final answer:
The percentage of values in the distribution between 14 and 26 in a normal distribution with a mean of 14 and a standard deviation of 4 is 97.65%, according to the 68-95-99.7 (empirical) rule.
Explanation:
According to the 68-95-99.7 rule (also known as the empirical rule), 68% of the data falls within one standard deviation of the mean, 95% percent within two standard deviations and 99.7% within three standard deviations in a normal distribution.
In your question, the mean is 14 and the standard deviation is 4. Therefore, values between 10 (14-4) and 18 (14+4) represent data within one standard deviation from the mean, values between 6 (14-2*4) and 22 (14+2*4) represent data within two standard deviations, and values between 2 (14-3*4) and 26 (14+3*4) represent data within three standard deviations.
From this, it's clear that the value 26 is a part of the range for three standard deviations, while 14 is the mean value. Thus, values between 14 and 26 make up half of the data in the two standard deviation range (2.5% to represent half of the 5% outside of the 95%) and half of the data in the three standard deviation range (0.15% to represent half of the 0.3% outside of the 99.7%). Therefore, to find the percentage of values between 14 and 26, you can simply add these percentages: 50% (half of one standard deviation: 68%) + 47.5% (half of two standard deviations: 95%) + 0.15% (half of three standard deviations: 99.7%) = 97.65%.
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