Respuesta :
Using the normal probability distribution and the central limit theorem, it is found that there is a 0.04 = 4% probability that the mean ticket price exceeds $35.
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Normal Probability Distribution
Problems of normal distributions can be solved using the z-score formula.
In a set 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]
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.
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Central Limit Theorem
The Central Limit Theorem establishes that, for a normally distributed random variable X, with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]s = \frac{\sigma}{\sqrt{n}}[/tex].
For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.
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The average ticket price for a Spring Training baseball game is $32.61, with a standard deviation of $8.62.
This means that [tex]\mu = 32.61, \sigma = 8.62[/tex]
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Sample of 40:
This means that [tex]n = 40, s = \frac{8.62}{\sqrt{40}}[/tex]
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In a random sample of 40 Spring Training tickets, find the probability that the mean ticket price exceeds $35.
This is 1 subtracted by the p-value of Z when X = 35, so:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
By the Central Limit Theorem
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{35 - 32.61}{\frac{8.62}{\sqrt{40}}}[/tex]
[tex]Z = 1.754[/tex]
[tex]Z = 1.754[/tex] has a p-value of 0.96.
1 - 0.96 = 0.04
0.04 = 4% probability that the mean ticket price exceeds $35.
A similar question is given at https://brainly.com/question/24342706
