The scores for all high school seniors taking the verbal section of the Scholastic Aptitude Test (SAT) in a particular year had a mean of 490 and a standard deviation of 100. The distribution of SAT scores is bell-shaped.

What percentage of seniors scored between 390 and 590 on this SAT test?

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okpalawalter8 okpalawalter8 · Answer 1 · 2020-11-29T03:03:57+01:00

Answer:

The value is [tex]P( 390 < X < 590) = 68.3 \%[/tex]

Step-by-step explanation:

From the question we are told that

The mean is [tex]\mu = 490[/tex]

The standard deviation is [tex]\sigma = 100[/tex]

Generally the proportion of seniors scored between 390 and 590 on this SAT test is mathematically represented as

[tex]P( 390 < X < 590) = P( \frac{390 - 490 }{100} < \frac{\= x - \mu }{\sigma} < \frac{590 - 490 }{ 100} )[/tex]

[tex]\frac{X -\mu}{\sigma } = Z (The \ standardized \ value\ of \ X )[/tex]

[tex]P( 390 < X < 590) = P( -1 < Z < 1 )[/tex]

=> [tex]P( 390 < X < 590) = P(Z< 1 ) - P(Z < - 1 )[/tex]

From the z table the area under the normal curve to the left corresponding to 1 and -1 is

[tex]P(Z< 1 ) = 0.84134[/tex]

and

[tex]P(Z< - 1 ) = 0.15866[/tex]

[tex]P( 390 < X < 590) =0.84134 - 0.15866[/tex]

=> [tex]P( 390 < X < 590) = 0.6827[/tex]

Generally the percentage of seniors scored between 390 and 590 on this SAT test is mathematically represented as

=> [tex]P( 390 < X < 590) = 0.6827 *100[/tex]

=> [tex]P( 390 < X < 590) = 68.3 \%[/tex]