Math / Data & Statistics

QuestionA certain standardized test's math scores have a bell-shaped distribution with a mean of 525 and a standard deviation of 110. Complete parts (a) through (c). (a) What percentage of standardized test scores is between 415 and 635? 68.0 \% (Round to one decimal place as needed.) (b) What percentage of standardized test scores is less than 415 or greater than 635 ? $\square$ \% (Round to one decimal place as needed.)

Studdy Solution

STEP 1

1. The distribution of test scores is normal (bell-shaped).
2. The mean ( $\mu$ ) is 525.
3. The standard deviation ( $\sigma$ ) is 110.
4. The empirical rule (68-95-99.7 rule) applies to normal distributions.

STEP 2

1. Use the empirical rule to determine the percentage of scores between 415 and 635.
2. Calculate the percentage of scores less than 415 or greater than 635.

STEP 3

Identify the z-scores for 415 and 635 using the formula:
$z = \frac{X - \mu}{\sigma}$
For $X = 415$ :
$z = \frac{415 - 525}{110} = \frac{-110}{110} = -1$
For $X = 635$ :
$z = \frac{635 - 525}{110} = \frac{110}{110} = 1$

STEP 4

According to the empirical rule, approximately 68% of the data falls within one standard deviation ( $-1 \leq z \leq 1$ ) of the mean.

STEP 5

The percentage of scores less than 415 or greater than 635 is the complement of the percentage found in Step 2.
$100\% - 68\% = 32\%$
The percentage of standardized test scores less than 415 or greater than 635 is:
$\boxed{32.0\%}$

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