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The mean song length on Jacks 3 minutes and 3 seconds , with a Standard deviation of 24 seconds. Find the probability that the next song to play is no more than 3 minutes and 45 seconds.

Respuesta :

MathPhys

Answer:

0.9599

Step-by-step explanation:

First, convert the times to seconds.

3 min 3 sec = 183 sec

3 min 45 sec = 225 sec

Next, find the z-score.

z = (x − μ) / σ

z = (225 − 183) / 24

z = 1.75

Next, use a calculator or z score table to find the probability.

P(Z < 1.75) = 0.9599

Using the normal distribution, it is found that there is a 0.959 = 95.99% probability that the next song to play is no more than 3 minutes and 45 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.

Considering the measures in seconds, the mean and the standard deviation are given, respectively, by: [tex]\mu = 183, \sigma = 24[/tex].

3 minutes and 45 seconds is equivalent to 225 minutes, hence the probability that the next song to play is no more than 3 minutes and 45 seconds is the p-value of Z when X = 225.

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

[tex]Z = \frac{225 - 183}{24}[/tex]

Z = 1.75

Z = 1.75 hs a p-value of 0.9599.

0.9599 = 95.99% probability that the next song to play is no more than 3 minutes and 45 seconds.

More can be learned about the normal distribution at https://brainly.com/question/24663213

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