Question

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.

Answer 1

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

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

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:

`X \sim N(5.75241,0.06281)`

Where
`\mu=5.75241` and
`\sigma=0.06281`

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

`z=(x-\mu)/(\sigma)`

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

`-2 = (X-5.75241)/(0.06281)`

And if we solve for X we got:

`X = 5.75241 -2*0.06281 =5.6121`

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

`2 = (X-5.75241)/(0.06281)`

And if we solve for X we got:

`X = 5.75241 +2*0.06281 =5.8886`

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