Respuesta :
Answer:
a) A z-score of 2.49 means that your grade was 2.49 standard deviations above the mean of all grades.
b) 1.39% of the class scored lower than your friend.
Step-by-step explanation:
Problems of normally distributed samples are solved using the z-score formula.
In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the zscore of a measure X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.
You got a score of 2.49 on an exam.
(a) Write a sentence explaining what that means
A z-score of 2.49 means that your grade was 2.49 standard deviations above the mean of all grades.
(b) Your friend got a z-score of negative 2−2. If the grades satisfy the Nearly Normal Condition, about what percent of the class scored lower than your friend?
This percentage is the pvalue of [tex]Z = -2.2[/tex]
So 0.0139 = 1.39% of the class scored lower than your friend.
Z-score is the measurement of how far is the score from the mean. Only 2.28% of the class is behind your friend.
What is Z-score?
A Z-score helps us to understand how far is the data from the mean. It is a measure of how many times the data is above or below the mean. It is given by the formula,
[tex]Z = \dfrac{X- \mu}{\sigma}[/tex]
Where Z is the Z-score,
X is the data point,
μ is the mean and σ is the standard variable.
The Z-score is the measurement of how far is the score from the mean in terms of standard deviation. therefore, if the z- score is 1, that means that the difference between the z-score and the mean is equal to the standard deviation.
A.) As given that your score in the exam is 2.49, which means that your score is 2.49 times the standard deviation ahead of the mean. therefore, you are 2.49 times ahead of an average kid in the class. The statement that can describe this situation is that 'The score of 2.49 shows that you are 2.49 times ahead of the mean of the class'.
B.) As given that your friend's score in the exam is -2, which means that your friend is 2 times the standard deviation behind the mean. therefore, your friend is 2 times behind an average kid in the class.
Now, in order to find the percentage of kids that are behind your friend simply find the value of the z-score from the z-table, and the value of -2 in the z-score is 0.0228, therefore, only 2.28% of the class is behind your friend.
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