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.

For this case we want this probability:

And we can use the z score formula given by:

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

Step-by-step 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.

So we can assume

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:

And we can use the z score formula given by:

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