zg683746
01.04.2020 •
Mathematics
Water permeability of concrete can be measured by letting water flow across the surface and determining the amount lost (in inches per hour). Suppose that the permeability index x for a randomly selected concrete specimen of a particular type is normally distributed with mean value 1000 and standard deviation 150.
(a) How likely is it that a single randomly selected specimen will have a permeability index between 550 and 1300? (Round your answer to four decimal places.)
P(550 < x < 1300) =
(b) If the permeability index is to be determined for each specimen in a random sample of size 10, how likely is it that the sample average permeability index will be between 900 and 1100? (Round your answers to four decimal places.)
P(900 < x < 1100) =
Between 850 and 1300?
P(850 < x < 1300) =
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Ответ:
a) P(550 < x < 1300) = 0.9759
b) P(900 < x < 1100) = 0.9652
P(850 < x < 1300) = 0.9992
Step-by-step explanation:
To solve this question, we need to understand the normal probability distribution and the central limit theorem.
Normal probability distribution
Problems of normally distributed samples are solved using the z-score formula.
In a set with mean and standard deviation , the zscore of a measure X is given by:
The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.
Central Limit Theorem
The Central Limit Theorem estabilishes that, for a normally distributed random variable X, with mean and standard deviation , the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean and standard deviation .
For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.
In this problem, we have that:
a) P(550 < x < 1300) =
pvalue of Z when X = 1300 subtracted by the pvalue of Z when X = 550.
X = 1300
has a pvalue of 0.9772
X = 550
has a pvalue of 0.0013
0.9772 - 0.0013 = 0.9759
P(550 < x < 1300) = 0.9759
b)
Now we have
P(900 < x < 1100) =
pvalue of Z when X = 1100 subtracted by the pvalue of Z when X = 900.
X = 1100
By the Central Limit Theorem
has a pvalue of 0.9826
X = 900
has a pvalue of 0.0174
0.9826 - 0.0174 = 0.9652
P(900 < x < 1100) = 0.9652
P(850 < x < 1300) =
pvalue of Z when X = 1300 subtracted by the pvalue of Z when X = 850. So
X = 1300
has a pvalue of 1
X = 850
has a pvalue of 0.0008
1 - 0.0008 = 0.9992
P(850 < x < 1300) = 0.9992
Ответ:
Music club last box
Reason: There are equal number if members
Physics club first box
Reason: There are more older than young people
Sports Club third box
Reason: There are more younger people than older people
History Club takes the second box
Step-by-step explanation:
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