The manufacturer of a certain type of new cell phone battery claims that the average life span of the batteries is 500 charges; that is, the battery can be charged at least 500 times before failing. To investigate the claim, a consumer group will select a random sample of cell phones with the new battery and use the phones through 500 charges of the battery.a. If the claim is true, what is P( X ≤ 36.7) ?b. Based on the answer to part (a), if the claim is true, is a sample mean lifetime of 36.7 hours unusually short?c. If the sample mean lifetime of the 100 batteries were 36.7 hours, would you find the manufacturer's claim to be plausible? Explain.d. If the claim is true, what is P( X ≤ 39.8)?e. Based on the answer to part (d), if the claim is true, is a sample mean lifetime of 39.8 hours unusually short?f. If the sample mean lifetime of the 100 batteries were 39.8 hours, would you find the manufacturer's claim to be plausible? Explain.

To investigate the claim, a consumer group will select a random sample of cell phones with the new battery and use the phones through 500 charges of the battery.

According to the question,

The sample size is n = 100,

From the central limit theorem,

[tex]\sigma_x^{2} = \dfrac{\sigma^{2} }{n}\\\\\sigma_x^{2} = \dfrac{5^{2} }{100} \\\\\sigma_x^{2} = \dfrac{25}{100}\\\\\sigma_x^{2} = 0.25\\\\X-N (40, \ 0.025)[/tex]

The z-score of 36.7 is ,

[tex]z = \dfrac{36.7-40}{\sqrt{0.25}} \\\\z = -6.6[/tex]

There is z-score table which would corresponds to small z,

Therefore,

[tex]P(X\leq 36.7) = P(X\leq -6.6) = 0[/tex]

A sample mean lifetime of 36.7 hours is,

[tex]P(X\leq 36.7) = 0[/tex]

Yes, a sample mean lifetime of 36.7 hours unusually short is 0.

The manufacturer claim is not plausible, because the supposed probability.

[tex]P(X\leq 36.7) = 0[/tex]

Z - value corresponding to 39.8 is;

[tex]z = \dfrac{39.8-40}{\sqrt{0.25}} \\\\z = -0.40\\[/tex]

From the z value ,

[tex]P(X\leq 39.8) = P(Z\leq -0.40) = 0.3446[/tex]

A sample mean lifetimes of 39.8 hours is not unusually to short as the corresponding probability is 0.3466 .

Suggesting that more than one third of the samples of size 100 will have 39.8 or less sample mean life time.

The manufacture claim is plausible [tex]P(X\leq 39.8) = 0.3446[/tex] .

The supposed probability suggestion that aa mean life time of 39.8 hours is not an short duration.

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