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A breakfast cereal producer makes its most popular product by combining just raisins and flakes in each box of cereal. The amounts of flakes in the boxes of this cereal are normally distributed with a mean of 370\,\text{g}370g370, start text, g, end text and a standard deviation of 24\,\text{g}24g24, start text, g, end text. The amounts of raisins are also normally distributed with a mean of 170\,\text{g}170g170, start text, g, end text and a standard deviation of 7\,\text{g}7g7, start text, g, end text.
Let T=T=T, equals the total amount of product in a randomly selected box, and assume that the amounts of flakes and raisins are independent of each other.
Find the probability that the total amount of product is less than 575\,\text{g}575g575, start text, g, end text.
You may round your answer to two decimal places.
P(T<575)\approx

Sagot :

Using the normal distribution, it is found that there is a 0.9192 = 91.92% probability that the total amount of product is less than 575 g.

In a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the z-score of a measure X is given by:

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

  • It 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, which is the percentile of X.
  • When two normal variables are added, the mean is the sum of the means while the standard deviation is the square root of the sum of the variances.

In this problem, the product is composed by flakes and raisins, and we have that:

[tex]\mu_F = 370, \sigma_F = 24, \mu_R = 170, \sigma_R = 7[/tex]

Hence, the distribution for the total amount of product has mean and standard deviation given by:

[tex]\mu = \mu_F + \mu_R = 370 + 170 = 540[/tex]

[tex]\sigma = \sqrt{\sigma_F^2 + \sigma_R^2} = \sqrt{24^2 + 7^2} = 25[/tex]

The probability that the total amount of product is less than 575 g is the p-value of Z when X = 575, hence:

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

[tex]Z = \frac{575 - 540}{25}[/tex]

[tex]Z = 1.4[/tex]

[tex]Z = 1.4[/tex] has a p-value of 0.9192.

0.9192 = 91.92% probability that the total amount of product is less than 575 g.

A similar problem is given at https://brainly.com/question/22934264

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