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Sagot :
Using the normal distribution and the central limit theorem, it is found that there is an approximately 0% probability that the total number of candies Kelly will receive this year is smaller than last year.
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, for n instances of a normal variable, the mean is [tex]n\mu[/tex] while the standard deviation is [tex]s = \sigma\sqrt{n}[/tex].
In this problem:
- Mean of 4 candies, hence [tex]\mu = 4[/tex].
- Standard deviation of 1.5 candies, hence [tex]\sigma = 1.5[/tex].
- She visited 35 houses, hence [tex]n = 35, \mu = 35(4) = 140, s = 1.5\sqrt{4} = 3[/tex]
The probability is the p-value of Z when X = 122, hence:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
By the Central Limit Theorem
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{122 - 140}{3}[/tex]
[tex]Z = -6[/tex]
[tex]Z = -6[/tex] has a p-value of 0.
Approximately 0% probability that the total number of candies Kelly will receive this year is smaller than last year.
A similar problem is given at https://brainly.com/question/24663213
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