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QuestionA scientist tests guava's effectiveness in filtering waste water. What is the probability that it absorbs more than 80ppm80 \mathrm{ppm}?

Studdy Solution

STEP 1

Assumptions1. The average absorption of waste water by the guava component is95 ppm. . The standard deviation of the absorption is15 ppm.
3. We assume a normal distribution of the absorption values.

STEP 2

First, we need to standardize the value of80 ppm to a z-score. The z-score is a measure of how many standard deviations an element is from the mean. The formula for the z-score isZ=X−μσZ = \frac{X - \mu}{\sigma}where- XX is the value we are interested in (80 ppm in this case), - μ\mu is the mean (95 ppm in this case), - σ\sigma is the standard deviation (15 ppm in this case).

STEP 3

Now, plug in the given values for XX, μ\mu, and σ\sigma to calculate the z-score.
Z=80−9515Z = \frac{80 -95}{15}

STEP 4

Calculate the z-score.
Z=80−9515=−1Z = \frac{80 -95}{15} = -1

STEP 5

The z-score of -1 means that80 ppm is one standard deviation below the mean. Now, we need to find the probability that a certain guava component absorbs more than80 ppm of waste water. This is equivalent to finding the probability that the z-score is greater than -1.Since the total area under the curve of a normal distribution is1, and the curve is symmetric, the area to the right of the mean (z=0) is0.5.

STEP 6

For a z-score of -1, the area to the left of it (which is the probability that the z-score is less than -1) is0.158 (you can find this value in a standard normal distribution table or use a calculator with a normal distribution function).

STEP 7

Since we are interested in the probability that the z-score is greater than -1 (which is the area to the right of z=-1), we subtract the area to the left of z=-1 from1.
(Z>−1)=1−(Z<−1)(Z > -1) =1 -(Z < -1)

STEP 8

Plug in the value for (Z<−1)(Z < -1) to calculate (Z>−1)(Z > -1).
(Z>−1)=1−0.1587(Z > -1) =1 -0.1587

STEP 9

Calculate the probability that a certain guava component absorbs more than80 ppm of waste water.
(Z>−)=−.1587=.8413(Z > -) = -.1587 =.8413So, the probability that a certain guava component absorbs more than80 ppm of waste water is.8413 or84.13%.

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