Question

Suppose that grade point averages of undergraduate students at one university have a bell-shaped distribution with a mean of 2.542.54 and a standard deviation of 0.420.42. Using the empirical rule, what percentage of the students have grade point averages that are between 1.281.28 and 3.83.8?

Answer 1

Answer:

$P(1.28< X< 3.8)$

And we can use the z score formula to calculate how many deviations we are within the mean

$z = \frac{X -\mu}{\sigma}$

And if we use this formula we got:

$z = \frac{1.28-2.54}{0.42}= -3$

$z = \frac{3.8-2.54}{0.42}= 3$

And using the empirical rule we know that within 3 deviation from the mean we have 99.7% of the values

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 grade point averages of undergraduate students.

From the problem we have the mean and the standard deviation for the random variable X. $E(X)=2.54, Sd(X)=0.42$

So we can assume $\mu=2.54 , \sigma=0.42$

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

For this case we want to find this probability:

$P(1.28< X< 3.8)$

And we can use the z score formula to calculate how many deviations we are within the mean

$z = \frac{X -\mu}{\sigma}$

And if we use this formula we got:

$z = \frac{1.28-2.54}{0.42}= -3$

$z = \frac{3.8-2.54}{0.42}= 3$

And using the empirical rule we know that within 3 deviation from the mean we have 99.7% of the values

Answer 2

Answer:

By the Empirical Rule, 99.7% of the students have grade point averages that are between 1.28 and 3.8.

Step-by-step explanation:

The Empirical Rule states that, for a normally distributed(bell-shaped) random variable:

68% of the measures are within 1 standard deviation of the mean.

95% of the measures are within 2 standard deviation of the mean.

99.7% of the measures are within 3 standard deviations of the mean.

In this problem, we have that:

Mean = 2.54

Standard deviation = 0.42.

Between 1.28 and 3.8?

1.28 = 2.54 - 3*0.42

So 1.28 is 3 standard deviations below the mean

3.8 = 2.54 + 3*0.42

So 3.8 is 3 standard deviations above the mean

By the Empirical Rule, 99.7% of the students have grade point averages that are between 1.28 and 3.8.