A laboratory weighs filters from a coal mine to measure the amount of dust in the mine atmosphere. Repeated measurements of the weight of dust on the same filter vary normally with standard deviation  0.08 milligram (mg) because the weighing is not perfectly precise. The dust on a particular filter actually weighs 123 mg. Repeated weighings will then have the N(123, 0.08) distribution. a) The lab reports the mean of 3 weighings. What is the distribution of this mean? b) What is the probability that the lab reports a result of 124 mg or higher for this filter?

a)

b)

And using a calculator, excel or the normal standard table we have that:

Step-by-step explanation:

Previous concepts

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

The Z-score is "a numerical measurement used in statistics of a value's relationship to the mean (average) of a group of values, measured in terms of standard deviations from the mean".  

Solution to the problem

Part a

Let X the random variable that represent the weights of a population, and for this case we know the distribution for X is given by:

Where and

Since the distribution for X is normal then we know that the distribution for the sample mean is given by:

Part b

We want to find this probability:

And we can use the z score given by:

And using this formula we got:

And using a calculator, excel or the normal standard table we have that:

step-by-step explanation:

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