Using cumulative density function of normal distribution, we find that out of 120 nails with a mean length of 5.00 inches and standard deviation of 0.03 inches, approximately 3 nails are less than 4.94 inches.
What is cumulative density function?
Cumulative density function gives the probability of finding the random variable at a value less than or equal to a given cutoff, i.e., P(X ≤ x).
To find the CDF of X ∼ N (μ,σ²), we can write
[tex]F_{X} (x) =P(X < =x) = P(Z < = \frac{x - \mu}{\sigma} )\\[/tex]
=Φ((x−μ)/σ)
Given,
x = 4.94
μ = 5
σ = 0.03
P(X <= 4.94) = Φ[tex](\frac{4.94-5}{0.03})[/tex] = Ф(-2) = 0.022750132 (using excel function norm.s.dist)
2.27% nails are less than 4.94 inches long.
To calculate the number of nails in sample of 120 that are less than 4.94 =
2.27% of 120 = 2.73 ~ 3
Learn more about cumulative density function here
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