Respuesta :
Answer:
[tex] z= \frac{3.44-2.54}{0.45}=2[/tex]
[tex] P(X>3.44) = P(Z>2) = 0.025[/tex]
Step-by-step 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 grade points avergae of a population, and for this case we know the following properties
Where [tex]\mu=2.54[/tex] and [tex]\sigma=0.45[/tex]
The empirical rule, also referred to as the 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). Broken down, the empirical rule shows that 68% falls within the first standard deviation (µ ± σ), 95% within the first two standard deviations (µ ± 2σ), and 99.7% within the first three standard deviations (µ ± 3σ).
So we can find the z score for the value of X=3.44 in order to see how many deviations above or belowe we are from the mean like this:
[tex] z= \frac{3.44-2.54}{0.45}=2[/tex]
So the value of 3.44 is 2 deviations above from the mean, so then we know that the percentage between two deviations from the mean is 95% and on each tail we need to have (100-95)/2 = 2.5% , because the distribution is symmetrical, so based on this we can conclude that:
[tex] P(X>3.44) = P(Z>2) = 0.025[/tex]
