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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 65 and 12, respectively. If the scores are normally distributed, what percentage

In a statistics class, the mean and standard deviation of the scores obtained by students on a test were found to be 65 and 12, respectively. If the scores are normally distributed, what percentage of students scored below 70 ?

Option: 1

52.22
 


Option: 2

96.14
 


Option: 3

89.63
 


Option: 4

66.32


Answers (1)

best_answer

To determine the percentage of students who scored below 70 , we can use the properties of the normal distribution.

First, we need to calculate the Z-score for a score of 70 . The Z-score represents the number of standard deviations a particular value is from the mean.
The formula to calculate the Z-score is:

\mathrm{ Z=(X-\mu) / \sigma }
Where:

\mathrm{ \begin{aligned} Z & =Z-\text { score } \\ X & =\text { Score } \\ \mu & =\text { Mean } \\ \sigma & =\text { Standard deviation } \end{aligned} }

Plugging in the values:

\mathrm{ \begin{aligned} & Z=(70-65) / 12 \\ & Z=5 / 12 \\ & Z \approx 0.4167 \end{aligned} }

Next, we need to find the cumulative probability associated with the Z-score of 0.4167 . This represents the percentage of values below 70 .

Using a standard normal distribution table or a calculator, we can find that the cumulative probability associated with a Z-score of 0.4167 is approximately 0.6632 .
To convert this probability to a percentage, we multiply it by 100 :

\mathrm{ \begin{aligned} & \text { Percentage }=0.6632 \times 100 \\ & \text { Percentage } \approx 66.32 \% \end{aligned} }
Therefore, approximately 66.32 \% of students scored below 70 on the test.

Posted by

Kuldeep Maurya

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