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  • User In a statistics class, the mean and standard deviation of the scores obtained by students on a test were found to be 75 and 9 , respectively. If the scores are normally distributed, wha

User
In a statistics class, the mean and standard deviation of the scores obtained by students on a test were found to be 75 and 9 , respectively. If the scores are normally distributed, what percentage of students scored below 70 ?

Option: 1

52.33 \%


Option: 2

28.93 \%


Option: 3

78.25 \%


Option: 4

63.24 \%


Answers (1)

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=(70-75) / 9 \\ & Z=-5 / 9 \\ & Z \approx-0.5556 \end{aligned} }

Next, we need to find the cumulative probability associated with the Z-score of -0.5556 . 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 -0.5556 is approximately 0.2893 .

To convert this probability to a percentage, we multiply it by 100 :

\mathrm{ \begin{aligned} \text { Percentage } & =0.2893 \times 100 \\ \text { Percentage } & \approx 28.93 \% \end{aligned} }
Therefore, approximately 28.93 \% of students scored below 70 on the test.

Posted by

Ramraj Saini

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