Question

Foot length in millimeters for a sample of 2000 babies is approximately normally distributed with a mean of 81.0. If the standard deviation is 5.0 mm, how many babies would the empirical rule suggest have feet of length longer than 86.0 mm but shorter than 96.0 mm?

Answer 1

Empirical Rule: The Empirical Rule states that almost all data lies within 3 standard deviations of the mean for a normal distribution.

Under this rule, 68% of the data falls within one standard deviation, 95% of the data lies within two standard deviations and 99.7% of the data falls within three standard deviation.

Now, X be a continuous random variable representing the foot length in millimeters.

Here, N(sample size) = 1000

(mean) = 81

(standard deviation) = 5

Now X ~ N( = 81, = 5²)

We need to find the number of babies with a foot length between 86.0 mm and 96.0 mm.

In order to find that we will first find the probability of babies having a foot length between 86.0 mm and 96.0 mm.

Hence, we need to find P(86 < x < 96).

We know that, z =

P(86 < x < 96) = = P(1 < z < 3) = P(0 < z < 3) - P(0 < z < 1)

P(86 < x < 96) = 0.4987 - 0.3413 = 0.1574

Now the percentage to babies with the foot length between 86.0 mm and 96.0 mm is given by multiplying

P(86 < x < 96) by 100.

P(86 < x < 96) 100 = 0.1574 100 = 15.74%

Now to get the number of babies that have a foot length between 86.0 mm and 96.0 mm is given by taking 15.74% of the total number of babies.

Total number of babies = 15.74% N = 0.1574 2000 = 314.8 315

Hence, there are 315 babies that have a feet of length longer than 86.0 mm but shorter than 96.0 mm.

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