Respuesta :
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.
Answer:
[tex] P(1.28< X< 3.8) [/tex]
And we can use the z score formula to calculate how many deviations we are within the mean
[tex] z = \frac{X -\mu}{\sigma}[/tex]
And if we use this formula we got:
[tex] z = \frac{1.28-2.54}{0.42}= -3[/tex]
[tex] z = \frac{3.8-2.54}{0.42}= 3[/tex]
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. [tex]E(X)=2.54, Sd(X)=0.42[/tex]
So we can assume [tex]\mu=2.54 , \sigma=0.42[/tex]
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:
[tex] P(1.28< X< 3.8) [/tex]
And we can use the z score formula to calculate how many deviations we are within the mean
[tex] z = \frac{X -\mu}{\sigma}[/tex]
And if we use this formula we got:
[tex] z = \frac{1.28-2.54}{0.42}= -3[/tex]
[tex] z = \frac{3.8-2.54}{0.42}= 3[/tex]
And using the empirical rule we know that within 3 deviation from the mean we have 99.7% of the values
