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The manufacturer of a certain brand of aluminum foil claims that the amount of foil on each roll follows a Normal distribution with a mean of 250 square feet (ft^2) and a standard deviation of 2 ft^2 . To test this claim, a restaurant randomly selects 10 rolls of this aluminum foil and carefully measures the mean area to be ¯ = 249.6 ft2 .

Find the probability that the sample mean area is 249.6 ft^2 or less if the manufacturer’s claim is true.


0.4207


0.0228


0.5793


0.7364


0.2636

Respuesta :

JeanaShupp

Answer: 0.4207

Step-by-step explanation:

Let [tex]\overline{x}[/tex] be the sample mean area.

Given: Population mean : [tex]\mu=250[/tex] sq. feet

Standard deviation: [tex]\sigma=2[/tex] sq. feet

Sample size : n= 10

The probability that the sample mean area is [tex]249.6 \text{ ft}^2[/tex] or less if the manufacturer’s claim is true.

[tex]P(\overline{x}<249.6)=P(\dfrac{\overline{x}-\mu}{\sigma}<\dfrac{249.6-250}{2})\\\\= P(z<-0.2)\ \ \ \ [z=\dfrac{\overline{x}-\mu}{\sigma}]\\\\=1-P(z<0.2)\\=1- 0.5793\\\\=0.4207[/tex]

Required probability = 0.4207

Using the normal distribution and the central limit theorem, it is found that the probability that the sample mean area is 249.6 ft^2 or less if the manufacturer’s claim is true is of 0.2636.

Normal Probability Distribution

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.
  • By the Central Limit Theorem, the sampling distribution of sample means of size n has standard deviation [tex]s = \frac{\sigma}{\sqrt{n}}[/tex].

In this problem:

  • The mean is of 250 ft², hence [tex]\mu = 250[/tex].
  • The standard deviation is of 2 ft², hence [tex]\sigma = 2[/tex].
  • A sample of 10 rolls is taken, hence [tex]n = 10, s = \frac{2}{\sqrt{10}}[/tex].

The probability that the sample mean area is 249.6 ft^2 or less if the manufacturer’s claim is true is the p-value of Z when X = 249.6, hence:

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

By the Central Limit Theorem

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

[tex]Z = \frac{249.6 - 250}{\frac{2}{\sqrt{10}}}[/tex]

[tex]Z = -0.6325[/tex]

[tex]Z = -0.6325[/tex] has a p-value of 0.2636, which is the probability.

To learn more about the normal distribution and the central limit theorem, you can take a look at https://brainly.com/question/24663213

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