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  • In a statistics class, the mean and standard deviation of the scores obtained by students on a test were found to be 73 and 20 , respectively. If the scores are normally distributed, what percentag

In a statistics class, the mean and standard deviation of the scores obtained by students on a test were found to be 73 and 20 , respectively. If the scores are normally distributed, what percentage of students scored below 78 ?

Option: 1

78.96 \%


Option: 2

46.45 \%


Option: 3

96.75 \%


Option: 4

59.87 \%


Answers (1)

best_answer

To determine the percentage of students who scored below 78 , we need to calculate the z-score corresponding to that value and then find the area under the normal curve to the left of that z-score.

The z-score is calculated using the formula:

\mathrm{ z=(x-\mu) / \sigma }
where:
x is the value we want to find the percentage below (in this case, 78 ),

\mu is the mean (73), and

\sigma is the standard deviation (20).

Plugging in the values:

\mathrm{ \begin{aligned} & z=(78-73) / 20 \\ & z=5 / 20 \\ & z=0.25 \end{aligned} }

Now we need to find the area to the left of this z-score in the standard normal distribution table (also known as the z-table) or by using a statistical software. The area represents the percentage of students who scored below 78 .

Looking up the z-score of 0.25 in the z-table, we find that the area to the left is approximately 0.5987 .

So, approximately 59.87% of the students scored below 78 on the test.

 

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Anam Khan

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