banner

The probability density function of the net weight in pounds of a packaged chemical herbicide is f(x)=2.2 for 49.8 < x < 50.2 pounds. a) Determine the probability that a package weighs more than 50 pounds. b) How much chemical is contained in 90% of all packages?

The question aims to find the probability that a package can weigh more than 50 pounds and how much chemical is contained in 90% of the package.

The question depends on the concept of Probability Density Function (PDF). The PDF is the probability function that represents the likelihood of all the values of the continuous random variable.

A probability density function or PDF is used in probability theory to describe the chance of a random variable staying within a particular specific range of values. These functions describe the probability density function of normal distribution and how there exists mean and deviation.

Expert Answer

The probability density function of the net weight in pounds for all the packaged chemical herbicides is given as:

\[ f(x) = 2.2 \hspace{0.2in} 49.8 \lt x \lt 50.2\ lbs \]

a) To calculate the probability that a package of chemical herbicides will weigh more than 50 pounds, we can integrate the probability density function. It is given as:

\[ P ( X \gt 50 ) = \int_{50}^{50.2} 2.2 \, dx \]

\[ P ( X \gt 50 ) = 2.2 \big[ x \big]_{50}^{50.2} \]

\[ P ( X \gt 50 ) = 2.2 \big[ 50.2\ -\ 50 \big] \]

\[ P ( X \gt 50 ) = 2.2 \times 0.2 \]

\[ P ( X \gt 50 ) = 0.44 \]

b) To calculate how much chemical is contained in 90% of all the packages of herbicide, let us use the same formula as above. The only difference from the above equation is that we have the final probability. We need to find the chemical quantity that yields that probability. The equation is given as:

\[ P ( X \gt x ) = \int_{x}^{50.2} 2.2 \, dx \]

\[ P ( X \gt x ) = 2.2 \big[ x \big]_{x}^{50.2} \]

\[ P ( X \gt x ) = 2.2 \big[ 50.2\ -\ x \big] \]

\[ P ( X \gt x ) = 110.44\ -\ 2.2x \]

\[ 0.90 = 110.44\ -\ 2.2x \]

\[ x = \dfrac{ 110.44\ -\ 0.90 }{ 2.2 } \]

\[ x = 49.79 \]

Numerical Result

a) The probability that a package of chemical herbicide will weigh more than 50 pounds is calculated to be:

\[ P ( X \gt 50 ) = 0.44 \]

b) The chemical in 90% of all the packages of herbicide is calculated to be:

\[ x = 49.79 \]

Example

The probability density function of package weight in kilograms is given below. Find the probability that will weigh more than 10 kg.

\[ f(x) = 1.7 \hspace{0.3in} 9.8 \lt x \lt 10.27 kg \]

The probability that a package will weigh more than 10 kg is given as:

\[ P ( X \gt 10 ) = \int_{10}^{10.27} 1.7 \, dx \]

\[ P ( X \gt 10 ) = 1.7 \big[ x \big]_{10}^{10.27} \]

\[ P ( X \gt 10 ) = 1.7 \big[ 10.27\ -\ 10 \big] \]

\[ P ( X \gt 10 ) = 1.7 \times 0.27 \]

\[ P ( X \gt 10 ) = 0.459 \]

5/5 - (5 votes)