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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 69 and 14, 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 69 and 14, respectively. If the scores are normally distributed, what percentage of students scored below 74 ?

Option: 1

22.52 \%


Option: 2

78.23 \%


Option: 3

64.49 \%


Option: 4

44.68 \%


Answers (1)

best_answer

To determine the percentage of students who scored below 74 , we can use the properties of the normal distribution.

First, we need to calculate the Z-score for a score of 74 . 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=(74-69) / 14 \\ & Z=5 / 14 \\ & Z \approx 0.3571 \end{aligned} }

Next, we need to find the cumulative probability associated with the Z-score of 0.3571 . This represents the percentage of values below 74 .

Using a standard normal distribution table or a calculator, we can find that the cumulative probability associated with a Z-score of 0.3571 is approximately 0.6449 .

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

\mathrm{ \begin{aligned} & \text { Percentage }=0.6449 \times 100 \\ & \text { Percentage } \approx 64.49 \% \end{aligned} }

Therefore, approximately 64.49 \% of students scored below 74 on the test.

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