PLEASE PAY SPECIAL ATTENTION TO THE AMOUNT OF DECIMAL PLACES ANSWERS NEED TO BE

Question

PLEASE PAY SPECIAL ATTENTION TO THE AMOUNT OF DECIMAL PLACES ANSWERS NEED TO BE ROUNDED TO.

Power +, Inc. produces AA batteries used in remote-controlled toy cars. The mean life of these batteries follows the normal probability distribution with a mean of 37 hours. hours and a standard deviation of 5.1 hours. As a part of its quality assurance program, Power +, Inc. tests samples of 9 batteries.

What is the standard error of the distribution of the sample mean? (Round your answer to 4 decimal places.)

What proportion of the samples will have a mean useful life of more than 38 hours? (Round z value to 2 decimal places and final answer to 4 decimal places.)

What proportion of the sample will have a mean useful life greater than 35.5 hours? (Round z value to 2 decimal places and final answer to 4 decimal places.)

What proportion of the sample will have a mean useful life between 35.5 and 38 hours? (Round z value to 2 decimal places and final answer to 4 decimal places.)

PLEASE PAY SPECIAL ATTENTION TO THE AMOUNT OF DECIMAL PLACES ANSWERS NEED TO BE ROUNDED TO.

Power +, Inc. produces AA batteries used in remote-controlled toy cars. The mean life of these batteries follows the normal probability distribution with a mean of 37 hours. hours and a standard deviation of 5.1 hours. As a part of its quality assurance program, Power +, Inc. tests samples of 9 batteries.

Explanation / Answer

Power +, Inc. produces AA batteries used in remote-controlled toy cars.

The mean life of these batteries follows the normal probability distribution with a mean (mu) of 37 hours. hours and

a standard deviation of (sd) 5.1 hours.

number of batteries (n) = 9

What can you say about the shape of the distribution of the sample mean?

The shape of the distribution of the sample mean is approximately normal with mean is population mean and standard deviation is sd/sqrt(n) (standard error).

What is the standard error of the distribution of the sample mean?

Standard error (SE) = sd / sqrt(n) = 1.7

What proportion of the samples will have a mean useful life of more than 38 hours?

That is here we have to calculate P(Xbar > 38).

Convert Xbar into z-score.

z = (Xbar - mu) / se

z = (38 - 37) / 1.7 = 0.59

Now we have to calculate P(Z > 0.59)

EXCEL syntax for finding this probability is,

=1 - NORMSDIST(z) (The probability is right sided)

z is the test statistic value.

P(Z > 0.59) = 0.2782

The proportion of the samples will have a mean useful life of more than 38 hours is 0.2782*100 = 27.82%

What proportion of the sample will have a mean useful life greater than 35.5 hours?

That is here we have to calculate P(Xbar > 35.5).

Convert Xbar into z-score.

z = (Xbar - mu) / se

z = (35.5 - 37) / 1.7 = -0.88

Now we have to calculate P(Z > -0.88)

EXCEL syntax for finding this probability is,

=1 - NORMSDIST(z) (The probability is right sided)

z is the test statistic value.

P(Z > -0.88) = 0.8112

The proportion of the samples will have a mean useful life of more than 35.5 hours is 0.8112*100 = 81.12%

What proportion of the sample will have a mean useful life between 35.5 and 38 hours?

That is here we have to calculate P(35.5 < Xbar > 38).

Convert Xbar into z-score.

z = (Xbar - mu) / se

z-score for 35.5 and 38 is,

z = (35.5 - 37) / 1.7 = -0.88

z = (38 - 37) / 1.7 = 0.59

Now we have to calculate P(-0.88 < Z < 0.59) = P(Z < 0.59) - P(Z < -0.88)

EXCEL syntax,

=NORMSDIST(z)

z is the test statistic value.

P(-0.88 < Z < 0.59) = 0.7218 - 0.1888 = 0.5330

The proportion of the sample will have a mean useful life between 35.5 and 38 hours is 0.5330*100 = 53.30%

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