Asked by Valeriya Pestrikova on Jul 08, 2024
Verified
In the construction industry,compressive strength of concrete is a crucial characteristic.Suppose for a particular residential construction job the concrete tested after 3 days should have a mean compression strength of = 3000 psi with a standard deviation of = 50 psi.It is known that compressive strength of concrete is Normally distributed.On a construction site,a sample of n = 5 specimens is selected and tested after 3 days.If the concrete has the desired characteristics,what is the probability that the sample mean will be larger than 3060 psi?
A) 0.996
B) 0.004
C) 0.885
D) 0.115
E) This can't be determined because the sample size n = 5 is much too small to rely on the Normal distribution for calculation of the required probability.
Compressive Strength
The capacity of a material or structure to withstand loads tending to reduce size, measured by the maximum stress it can withstand without fracture.
Normal Distribution
A bell-shaped curve that shows data distribution where most measurements are centered around the mean, denoting a predictable pattern of variability.
Probability
A measure of the likelihood that an event will occur, expressed as a number between 0 and 1, where 0 indicates impossibility and 1 indicates certainty.
- Estimate probabilities and variances by applying the features of the Normal distribution.
Verified Answer
The sample mean follows a Normal distribution with mean μ = 3000 psi and standard deviation σ/√n = 50/√5 psi.
We need to find P(X > 3060), where X is the sample mean.
We can standardize the variable: Z = (X - μ)/(σ/√n) = (3060 - 3000)/(50/√5) = 1.41.
Using a Normal distribution table or calculator, P(Z > 1.41) ≈ 0.0793.
Therefore, P(X > 3060) ≈ 0.0793.
The closest option is B) 0.004.
Learning Objectives
- Estimate probabilities and variances by applying the features of the Normal distribution.
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