A standardized test is designed so that scores have a mean of 50 and a standard deviation of 4. What percent of scores are between 42 and 58?A.100%B. about 95.4%C. about 23.85%D. about 47.7%

Respuesta :

Step 1. The information that we have is:

The mean:

[tex]\mu=50[/tex]

The standard deviation:

[tex]\sigma=4[/tex]

Step 2. To solve this problem and find what percent of scores are between 42 and 58, we use the empirical rule:

• The empirical rule for normally distributed data tells us that about 68% of the data falls under 1 standard deviation from the mean, about 95% falls under 2 standard deviations from the mean, and 99.7% of the data falls under 3 standard deviations from the mean.

Step 3. The following diagram represents the situation:

The marks on the graph are calculated as follows:

[tex]\begin{gathered} \mu-\sigma=50-4=46 \\ \mu+\sigma=50+4=54 \\ \mu+2\sigma=50-2\cdot4=50-8=58 \\ \mu-2\sigma=50-2\times4=50-8=42 \end{gathered}[/tex]

This is represented in the image:

Step 4. As you can see in the previous graph, 42 and 58 are 2 standard deviations away from the mean, this means that about 95% of the data will be between those values.

The option closest to 95% is B. about 95.4%

Answer: B. about 95.4%

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