Answer:
2.28% (2 d.p.)
Step-by-step explanation:
The shelf life of a particular dairy product is normally distributed with a mean (μ) of 12 days and a standard deviation (σ) of 3 days.
Therefore:
[tex]\rm X\sim N \left(\mu,\sigma^2\right)\implies \boxed{\rm X \sim N \left(12, 3^2\right)}[/tex]
where X is the shelf life in days.
To calculate the percentage of dairy products that have a shelf life of less than 6 days, we need to find P(X < 6), which is the same as P(X ≤ 6) since, in a continuous normal distribution, the probability of obtaining a specific value is zero.
Calculator input for "normal cumulative distribution function (cdf)":
- Lower bound: x = -100
- Upper bound: x = 6
- σ = 3
- μ = 12
This gives the percent of dairy products that have a shelf life of less that 6 days as:
P(X ≤ 6) = 0.02275013...
P(X ≤ 6) = 2.28% (2 d.p.)
To use the standard normal distribution (z-score) and then consult the z-tables, transform X to Z:
[tex]P(X \leq 6)=P\left(Z\leq \dfrac{6-12}{3}\right)=P(Z\leq -2)[/tex]
According to the z-tables, a z-score of -2 corresponds to approximately 0.0228 in the left tail.
Therefore, the percentage of the shelf life less than 6 days is approximately 2.28%.
