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Statistics Help

by Sarah
(Jacksonville, NC)











































Portland cement has an average strength of 4,000 pounds per square. This compressive
strength was proven to be normally distributed with a standard deviation of 120 psi.
Determine the probability that the compressive strength of the concrete will be more than
3,850 psi.

Comments for Statistics Help

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Oct 27, 2010
Application of Normal Distribution Statistics
by: Staff

The question:

by Sarah
(Jacksonville, NC)


Portland cement has an average strength of 4,000 pounds per square inch. This compressive strength was proven to be normally distributed with a standard deviation of 120 psi. Determine the probability that the compressive strength of the concrete will be more than 3,850 psi.



The answer:

I am going to a z-table to obtain an exact answer.

But first, we can estimate an approximate value by applying the 68-95-99.7% rule (this is also called the Empirical Rule).

The 68-95-99.7 rule states that plus or minus one standard deviation from the mean includes 68% of the normal distribution. Plus or minus two standard deviations from the mean includes 95% of the normal distribution. Plus or minus three standard deviations from the mean includes 99.7% of the normal distribution.

Almost all (99.7%) of the normal distribution data occurs within 3 standard deviations from the mean.

For this problem:
Your mean is: 4,000 psi
Your standard deviation is: 120 psi

One standard deviation below 4,000 psi is:
4000 - 120 = 3880 psi
Two standard deviations below 4,000 psi is:
4000 - 240 = 3760 psi

An estimate of the probability of strength of the concrete being above 3850 psi is:
Between 2 to 3 standard deviations below the mean of 4000 psi.

There is a 50% probability that the strength of the cement will be above the mean.

Because of this, we know that the probability of the strength exceeding 3850 psi is somewhere between 84% and 98.25%.

84% is one standard deviation below the mean (50 + 34 = 84%). 98.25% is two standard deviations below the mean (50 + 48.25 = 98.25% ).

To compute an exact answer, we will use a z-table.

Compute the Z value.

The question is: Exactly how many standard deviations from the mean is 3850 psi?

Z = (mean ? value)/(standard deviation)
Z = (4000 psi ? 3850 psi)/(120 psi)
Z = 1.25 standard deviations from the mean

The cumulative probability from the z-table is: 89.44%

Final answer: The probability that the compressive strength of the cement will be greater than 3850 psi is 89.44%
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Z table

Source: http://www.sjsu.edu/faculty/gerstman/EpiInfo/z-table.htm

The values in the body of this Z table represent the area under the normal distribution curve from the starting point and upwards. The starting point for this problem is a standard deviation of -1.25. (This is the Z value computed above.)

Z ::: 0.00::: 0.01:: 0.02::: 0.03:: 0.04::: 0.05:: 0.06::: 0.07:: 0.08::: 0.09

1.1 0.8643 0.8665 0.8686 0.8708 0.8729 0.8749 0.8770 0.8790 0.8810 0.8830
1.2 0.8849 0.8869 0.8888 0.8907 0.8925 0.8944 0.8962 0.8980 0.8997 0.9015
1.3 0.9032 0.9049 0.9066 0.9082 0.9099 0.9115 0.9131 0.9147 0.9162 0.9177
--------------------------------------------------------------------------------------





Thanks for writing.


Staff
www.solving-math-problems.com



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