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WITCHER [35]
2 years ago
11

A certain brand of candies have a mean weight of 0.8616g and a standard deviation of 0.0518 based on the sample of a package con

taining 447 candies. The package label stated that the net weight is 381.8g If every package has 447 candies the mean weight of the candies must exceed 381.8/447=0.8542 for the net contents to weight at least 381.8g.
a) If 1 candy is reandomly selected, find the probability that it weights more than 0.8542g.
The probability is ________.
(round to four decimal places as needed)
b) If 447 candies are reandomly selected find the probability that their mean weight is at least 0.8542 g.
the probability that a sample of 447 candies will have a mean of 0.8542g or greater is __________.
(round to four decimal places as needed)
c) Given these results does it seem that the candy company is providing consumers with the amount claimed on the label?
Mathematics
1 answer:
torisob [31]2 years ago
6 0

Answer:

a) The probability is 0.5557 = 55.57%.

b) The probability that a sample of 447 candies will have a mean of 0.8542g or greater is 0.9987 = 99.87%.

c) Yes, because there is a very large probability, of 99.87%, that the amount will be at least the one claimed on the label.

Step-by-step explanation:

To solve this question, we need to understand the normal probability distribution and the central limit theorem.

Normal Probability Distribution:

Problems of normal distributions can be solved using the z-score formula.

In a set with mean \mu and standard deviation \sigma, the z-score of a measure X is given by:

Z = \frac{X - \mu}{\sigma}

The Z-score 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. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the p-value, we get the probability that the value of the measure is greater than X.

Central Limit Theorem

The Central Limit Theorem estabilishes that, for a normally distributed random variable X, with mean \mu and standard deviation \sigma, the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean \mu and standard deviation s = \frac{\sigma}{\sqrt{n}}.

For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.

Mean weight of 0.8616g and a standard deviation of 0.0518

This means that \mu = 0.8616, \sigma = 0.0518

a) If 1 candy is reandomly selected, find the probability that it weights more than 0.8542g.

This is 1 subtracted by the pvalue of Z when X = 0.8542. So

Z = \frac{X - \mu}{\sigma}

Z = \frac{0.8542 - 0.8616}{0.0518}

Z = -0.14

Z = -0.14 has a pvalue of 0.4443

1 - 0.4443 = 0.5557

The probability is 0.5557 = 55.57%.

b) If 447 candies are reandomly selected find the probability that their mean weight is at least 0.8542 g.

Sample of 447 means that n = 447, s = \frac{0.0518}{\sqrt{447}} = 0.00245

Z = \frac{X - \mu}{\sigma}

By the Central Limit Theorem

Z = \frac{X - \mu}{s}

Z = \frac{0.8542 - 0.8616}{0.0245}

Z = -3.02

Z = -3.02 has a pvalue of 0.0013

1 - 0.0013 = 0.9987

The probability that a sample of 447 candies will have a mean of 0.8542g or greater is 0.9987 = 99.87%.

c) Given these results does it seem that the candy company is providing consumers with the amount claimed on the label?

Yes, because there is a very large probability, of 99.87%, that the amount will be at least the one claimed on the label.

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prisoha [69]

Using the normal distribution, it is found that 63.18% of the area under the curve of the standard normal distribution is between z = − 0.9 z = - 0.9.

<h3>Normal Probability Distribution</h3>

The z-score of a measure X of a normally distributed variable with mean \mu and standard deviation \sigma is given by:

Z = \frac{X - \mu}{\sigma}

  • The z-score measures how many standard deviations the measure is above or below the mean.
  • Looking at the z-score table, the p-value associated with this z-score is found, which is the percentile of X.

The area within 0.9 standard deviations of the mean is the <u>p-value of Z = 0.9(0.8159) subtracted by the p-value of Z = -0.9(0.1841)</u>, hence:

0.8159 - 0.1841 = 0.6318 = 63.18%.

More can be learned about the normal distribution at brainly.com/question/4079902

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Look at the right-angled triangle ABC.
olga_2 [115]

Answer:

∠x = 90°

∠y = 58°

∠z = 32°

Step-by-step explanation:

he dimensions of the angles given are;

∠B = 32°

Whereby ΔABC is a right-angled triangle, and the square fits at angle A, we have;

∠A = 90°

∠B + ∠C = 90° which gives

32° + ∠C = 90°

∠C = 58°

∠x + Interior angle of the square = 180° (Sum of angles on a straight line)

∠x + 90° = 180°

∠x = 90°

∠x + ∠y + 32° = 180° (Sum of angles in a triangle)

90° + ∠y + 32° = 180°

∠y = 180 - 90° - 32° = 58°

∠y + ∠z + Interior angle of the square = 180° (Sum of angles on a straight line)

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