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In a normally distributed set with a mean of 20 and standard deviation of 3, find the probability that a randomly selected x-value is between 14 and 26.

A. 0.6827
B. 0.8664
C. 0.9545
D. 0.9974

User Compholio
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1 Answer

4 votes

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.

User Mohamed Bana
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