Answered

Westonci.ca makes finding answers easy, with a community of experts ready to provide you with the information you seek. Get the answers you need quickly and accurately from a dedicated community of experts on our Q&A platform. Discover in-depth answers to your questions from a wide network of professionals on our user-friendly Q&A platform.

Sofia's audio player has 15,000 songs. the play time for the songs is skewed to the right, with a mean of 255 seconds and a standard deviation of 30 seconds. part a: can you accurately calculate the probability that the mean play time is more than 260 seconds for an srs of 15 songs? explain. (4 points) part b: if you take a random sample of 40 songs instead of 15, explain how the central limit theorem allows you to find the probability that the mean play time is more than 260 seconds. calculate this probability and show your work. (6 points)

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

#SPJ1

Thanks for using our platform. We aim to provide accurate and up-to-date answers to all your queries. Come back soon. We appreciate your time. Please revisit us for more reliable answers to any questions you may have. Thank you for using Westonci.ca. Come back for more in-depth answers to all your queries.