Question

A certain standardized​ test's math scores have a​ bell-shaped distribution with a mean of 520 and a standard deviation of 110. Complete parts​ (a) through​ (c). ​
(a) What percentage of standardized test scores is between 190 and 850​? ____ ​(Round to one decimal place as​ needed.) ​

(b) What percentage of standardized test scores is less than 190 or greater than 850​? ____ ​(Round to one decimal place as​ needed.)

​(c) What percentage of standardized test scores is greater than 740​? ____ ​(Round to one decimal place as​ needed.)

Answer 1

Answer:

(a) 99.7%

(b) 0.3%

(c) 2.5%

Step-by-step explanation:

This problem doesnt require a z-table as long as you know the basic rule of 68, 95 and 99.7.

If you are +-1 standard deviation from the mean the percentage in that band correspond to the 68% of the values.

If you are +-2 standard deviation from the mean the percentage in that band correspond to the 95% of the values.

If you are +-3 standard deviation from the mean the percentage in that band correspond to the 99.7% of the values.

The formula to calculate z-score is:

$z=\frac{x-\mu}{\sigma}$

$x\ is \ the\ number\ evaluated\\\mu\ is\ the\ mean\\\sigma\ is \ the \standard\ deviation$

(a)

lower z = (190 - 520)/110 = -3

upper z = (850 - 520)/110 = 3

This mean that you are looking for a percentage between +-3 standard deviation, exactly 99.7%

(b)

Since the percentage between is 99.7%, the percentage less than -3 standard deviation or greater than +3 standard deviation is 100 - 99.7 = 0.3%

(c)

z of 740 = (740 - 520)/110 = 2

The percentage between +-2 is 95%, you are interested in the percentage greater than 2 and that is (100 - 95)/2 = 2.5%. You have to divide by 2 because 2.5% is less than -2 standard deviation and 2.5% is greater than 2 standard deviation, and you are only interested in the percetange greater than that