Question

Suppose 120 geology students measure the mass of an ore sample. Due to human error and limitations in the reliability of the​ balance, not all the readings are equal. The results are found to closely approximate a normal​ curve, with mean 81 g and standard deviation 3 g. Use the symmetry of the normal curve and the empirical rule as needed to estimate the number of students reporting readings between 75 g and 87 g.

Answer 1

For this case we want this probability:

`P( 75 <X< 87)`

And we can use the z score formula given by:

`z = (87-81)/(3) = 2`

`z = (75-81)/(3) = -2`

So we want the probability for the values within two deviations from the mean and from the empirical rule we know that we have approximately 0.95 or 95% of the values within two deviations from the true mean.

And since we have a total of 120 students we expect:

120*0.95 = 114 between 75g and 87 g

Explanation:

Previous concepts

The empirical rule, also known as three-sigma rule or 68-95-99.7 rule, "is a statistical rule which states that for a normal distribution, almost all data falls within three standard deviations (denoted by σ) of the mean (denoted by µ)".

Let X the random variable who represent the courtship time (minutes).

From the problem we have the mean and the standard deviation for the random variable X.
`E(X)=81, Sd(X)=3`

So we can assume
`\mu=81 , \sigma=3`

On this case in order to check if the random variable X follows a normal distribution we can use the empirical rule that states the following:

• The probability of obtain values within one deviation from the mean is 0.68

• The probability of obtain values within two deviation's from the mean is 0.95

• The probability of obtain values within three deviation's from the mean is 0.997

Solution to the problem

For this case we want this probability:

`P( 75 <X< 87)`

And we can use the z score formula given by:

`z = (87-81)/(3) = 2`

`z = (75-81)/(3) = -2`

So we want the probability for the values within two deviations from the mean and from the empirical rule we know that we have approximately 0.95 or 95% of the values within two deviations from the true mean.

And since we have a total of 120 students we expect:

120*0.95 = 114 between 75g and 87 g