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The lifetimes of 20,000 light bulbs are normally distributed. The mean lifetime is 230 days. The standard deviation is 40 days. Find the values defined by standard deviation in a normal distribution for 3 standard deviations.

Respuesta :

Using the normal distribution, the values within 3 standard deviations of the mean are given 110 to 350.

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.

The mean and the standard deviation are given, respectively, by:

[tex]\mu = 230, \sigma = 40[/tex].

The bounds of the values within 3 standard deviations of the mean are given by X when Z = -3 and X when Z = 3, hence:

Z = -3:

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

[tex]-3 = \frac{X - 230}{40}[/tex]

X - 230 = -3(40)

X = 110.

Z = 3:

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

[tex]3 = \frac{X - 230}{40}[/tex]

X - 230 = 3(40)

X = 350.

More can be learned about the normal distribution at https://brainly.com/question/4079902

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