Question

Suppose that the walking step lengths of adult males are normally distributed with a mean of 2.6 feet and a standard deviation of 0.2 feet. A sample of 53 men’s step lengths is taken.

Step 1 of 2 : Find the probability that an individual man’s step length is less than 2.4 feet. Round your answer to 4 decimal places, if necessary.

Answer 1

To find the probability that an individual man's step length is less than 2.4 feet given the normally distributed step length with a mean of 2.6 and a standard deviation of 0.2, we calculate the Z-score and find that there is approximately a 15.87% chance.

Step-by-step explanation:

The question asks us to find the probability that an individual man's step length is less than 2.4 feet, given a normal distribution with a mean of 2.6 feet and a standard deviation of 0.2 feet. To find this probability, we use the standard normal distribution (Z-distribution).

Step 1: Convert the step length of 2.4 feet to a Z-score. The formula to calculate the Z-score is:

Z = (X - μ) / σ

Where X is the value of interest (2.4 feet), μ is the mean (2.6 feet), and σ is the standard deviation (0.2 feet).

Z = (2.4 - 2.6) / 0.2 = -1

Step 2: Use standard normal distribution tables, a calculator, or software to find the probability corresponding to a Z-score of -1.

The probability (P) that a man's step length is less than 2.4 feet is roughly 0.1587.

Therefore, there is a 15.87% chance that an individual man's step length is less than 2.4 feet.

Answer 2

P[ X < 2,4 ] = 0,1587 or 15,87 %

Step-by-step explanation:

Normal Distribution N ( 2,6 , 0,2 )

mean = μ₀ = 2,6 ft and

standard dviation is

σ = 0,2 ft

Samle size 53 n = 53

2,4 - 2,6 = 0,2

That means the difference between 2,4 and the mean of the sample ( 2,6) is equal to one standar deviation. The empircal rule establishes that the interval [ μ₀ ± σ] contans 68,3 % of all values, by symmetry

μ₀ - σ = 68,3/2

μ₀ - σ = 34,15

Then the half of the bell shape curve is 0,5 and from 0,2 up to 0 ( the normalized μ₀ ) is 34,15 then dfference between

0,5 - 0,3415 = 0,1585 or 15,85% is the probability

Other procedure is:

P[ X < 2,4 ] = [X - μ₀ ] / σ

P[ X < 2,4 ] = 2,4 - 2,6 / 0,2

P[ X < 2,4 ] = - 0,2/0,2

P[ X < 2,4 ] = - 1

And fom z - table we find for z (score) = - 1

P[ X < 2,4 ] = 0,1587