Problem

In a certain distribution, the mean is 80 with a standard deviation of 3 . At least what fraction of the numbers are between the following pair of numbers?
74 and 86
At least $\square$ of the numbers in the distribution are between 74 and 86 .
(Type an integer or simplified fraction.)

Answer

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Answer

Final Answer: \(\boxed{\frac{95}{100}}\)

Steps

Step 1 :This is a statistics problem and it involves the concept of normal distribution. In a normal distribution, about 68% of values drawn from that distribution are within one standard deviation 'σ' away from the mean; about 95% of the values lie within two standard deviations; and about 99.7% are within three standard deviations. This is known as the 68-95-99.7 (empirical) rule, or the 3-sigma rule.

Step 2 :In this case, the mean is 80 and the standard deviation is 3. The numbers 74 and 86 are respectively 2 standard deviations below and above the mean.

Step 3 :Therefore, according to the 3-sigma rule, at least 95% of the numbers in the distribution are between 74 and 86.

Step 4 :Final Answer: \(\boxed{\frac{95}{100}}\)

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