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The diameters of apples from a certain farm follow the normal distribution with mean 4 inches and standard deviation 0.4 inch. Apples can be size-sorted by being made to roll over mesh screens. First the apples are rolled over a screen with mesh size 3.5 inches. This separates out all the apples with diameters less than 3.5 inches. Second, the remaining apples are rolled over a screen with mesh size 4.3 inches. Find the proportion of apples with diameters less than 3.5 inches.

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Answer:

The probability for the proportion of apples with diameters less than 3.5 inches is 0.1056.

Explanation:

For a normal distribution, z score is calculated using this formula:  (X-μ)/σx

Where Mean, μ = 4

Standard deviation, σ = 0.400

The probability for the proportion of apples with diameters less than 3.5 inches is expressed below:

Probability = P(X<3.5)

                  = P(Z<-1.25)

Probability  = 0.1056

Answer:

P = 10%

Explanation:

Let,

  • µ = mean of the population = 4 inches
  • σ = standard deviation = 0.4 inches
  • X= a random variable that gives apple’s diameter
  • P [X < 3.5] = P-value is the probability of apples which will be separated after passing through the mesh screen

After putting all the values in the formula, we will find the value of Z-score from the chart.

P [X < 3.5]      = P [ [tex]\frac{X-4}{0.4}[/tex] < [tex]\frac{3.5-4}{0.4}[/tex] ]

P [Z < -1.25]    = 0.09680

About 10% of the apples possess diameters less than 3.5 inches.

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