Consider a value to be significantly low if its z score less than or equal to minus−2 or consider a value to be significantly high if its z score is greater than or equal to 2. A data set lists weights (grams) of a type of coin. Those weights have a mean of 5.752415.75241 g and a standard deviation of 0.062810.06281 g. Identify the weights that are significantly low or significantly high.

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

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:

[tex]X \sim N(5.75241,0.06281)[/tex]

Where [tex]\mu=5.75241[/tex] and [tex]\sigma=0.06281[/tex]

And the best way to solve this problem is using the normal standard distribution and the z score given by:

[tex]z=\frac{x-\mu}{\sigma}[/tex]

For the case when z =-2 we can do this:

[tex] -2 = \frac{X-5.75241}{0.06281}[/tex]

And if we solve for X we got:

[tex] X = 5.75241 -2*0.06281 =5.6121[/tex]

And for the other case when Z=2 we have:

[tex] 2 = \frac{X-5.75241}{0.06281}[/tex]

And if we solve for X we got:

[tex] X = 5.75241 +2*0.06281 =5.8886[/tex]

If the weight is higher than 5.8886 gr would be considered significantly high

If the weight is lower than 5.6121 gr would be considered significantly low