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The average lifetime of smoke detectors that a company manufactures is 5 years, or 60 months, and the standard deviation is 8 months. Find the probability that a random sample of 34 smoke detectors will have a mean lifetime between 58 and 63 months. Assume that the sample is taken from a large population and the correction factor can be ignored. Round the final answer to at least four decimal places and intermediate z-value calculations to two decimal places.

Respuesta :

The probability that a random sample of 34 smoke detectors will have a mean lifetime between 58 and 63 months, using the normal distribution, is of:

0.9135 = 91.35%.

How to obtain probabilities using the normal distribution?

The z-score of a measure X of a variable that has mean symbolized by [tex]\mu[/tex] and standard deviation symbolized by [tex]\sigma[/tex] is obtained by the rule presented as follows:

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

  • The z-score represents how many standard deviations the measure X is above or below the mean of the distribution, depending if the obtained z-score is positive or negative.
  • Using the z-score table, the p-value associated with the calculated z-score is found, and it represents the percentile of the measure X in the distribution.
  • By the Central Limit Theorem, the sampling distribution of sample means of size n has standard deviation [tex]s = \frac{\sigma}{\sqrt{n}}[/tex].

The parameters in the context of this problem are given as follows:

[tex]\mu = 60, \sigma = 8, n = 34, s = \frac{8}{\sqrt{34}} = 1.37[/tex]

The probability that a random sample of 34 smoke detectors will have a mean lifetime between 58 and 63 months is the p-value of Z when X = 63 subtracted by the p-value of Z when X = 58, hence:

X = 63:

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

By the Central Limit Theorem

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

Z = (63 - 60)/1.37

Z = 2.19

Z = 2.19 has a p-value of 0.9857.

X = 58:

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

Z = (58 - 60)/1.37

Z = -1.46

Z = -1.46 has a p-value of 0.0722.

0.9857 - 0.0722 = 0.9135 = 91.35%.

More can be learned about the normal distribution at https://brainly.com/question/25800303

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