Final answer:
The probability that a randomly selected x-value is between 14 and 26 in a normally distributed set with a mean of 20 and a standard deviation of 3 is approximately 0.9545, represented by the area under the standard normal curve between Z-scores of -2 and 2.
Step-by-step explanation:
The subject of this question is Mathematics, and it is a typical problem addressed at the High School level, especially in statistics or probability courses. To calculate the probability that a randomly selected x-value falls between 14 and 26 in a normally distributed set with a mean of 20 and a standard deviation of 3, we'll use the properties of the standard normal distribution (or Z-score).
First, we calculate the Z-scores for the given x-values. The Z-score is found using the formula: Z = (X - mean) / standard deviation.
- For X = 14: Z = (14 - 20) / 3 = -2
- For X = 26: Z = (26 - 20) / 3 = 2
To find the probability for the Z-scores, we refer to the standard normal distribution table or use a calculator. The probability that a Z-score is between -2 and 2 is approximately 0.9545 (Option C), which represents the area under the curve between these two Z-scores.
In terms of the area percentages for other standard deviations:
- About 68% of the x values lie within one standard deviation of the mean.
- About 95% of the x values lie within two standard deviations of the mean.
- About 99.7% of the x values lie within three standard deviations of the mean.