Question

Suppose a set of test scores is appproximately bell-shaped, with a mean of 68 and a range of 54. (a) The minimum test score is approximately , and the maximum test score is approximately . (b) The standard deviation of the test scores is approximately g

Answer 1

Answer:

a.) Minimum test score = 41

    Maximum test score = 95

b.)An approximate value of the standard deviation is 9.

Step-by-step explanation:

The given set of test scores is approximately bell shaped.

The mean of the test scores, $\mu$ = 68

The range of the data set = 54.

a.) i.) Minimum test score  = $\mu - \frac{range}{2} = 68 - \frac{54}{2} = 68 - 27 = 41$

   ii.) The maximum test score = $\mu + \frac{range}{2} = 68 + \frac{54}{2} = 68 + 27 = 95$

b.) From the 68-95-99.7 rule, also known as the Empirical rule, tell us that an approximation of the range is 3 standard deviations either way from the mean. So from this rule we can say that the minimum can also be found from $\mu - 3\sigma$ and the maximum can also be approximated as $\mu + 3\sigma$.

From the above we can say that the range can be written as

   $3\sigma - (-3\sigma) = range$

∴   $6\sigma = range$

∴    $\sigma = \frac{range }{6} = \frac{54}{6} = 9$

Therefore an approximate value of the standard deviation is 9.