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Sagot :
Using the normal distribution and the central limit theorem, it is found that:
a) Since the distribution is skewed and the sample size is less than 30, the probability cannot be calculated.
b) There is a 0.1469 = 14.69% probability that the mean play time is more than 260 seconds.
Normal Probability Distribution
The z-score of a measure X of a normally distributed variable with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex] is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
- 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.
- By the Central Limit Theorem, the sampling distribution of sample means of size n has standard deviation [tex]s = \frac{\sigma}{\sqrt{n}}[/tex], as long as the underlying distribution is normal and the sample size is at least 30.
The mean and the standard deviation are given by, respectively:
[tex]\mu = 255, \sigma = 30[/tex]
In item a, according to the Central Limit Theorem, since the distribution is skewed and the sample size is less than 30, the probability cannot be calculated.
For item b, we have that n = 40 > 15, hence the standard error is given by:
[tex]s = \frac{30}{\sqrt{40}} = 4.74[/tex]
The probability is one subtracted by the p-value of Z when X = 260, hence:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
By the Central Limit Theorem
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{260 - 255}{4.74}[/tex]
Z = 1.05
Z = 1.05 has a p-value of 0.8531.
1 - 0.8531 = 0.1469.
There is a 0.1469 = 14.69% probability that the mean play time is more than 260 seconds.
More can be learned about the normal distribution and the central limit theorem at https://brainly.com/question/24663213
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