Questions: A certain standardized test's math scores have a bell-shaped distribution with a mean of 530 and a standard deviation of 119. Complete parts (a) through (c) (a) What percentage of standardized test scores is between 173 and 887? 99.7 % (Round to one decimal place as needed.) (b) What percentage of standardized test scores is less than 173 or greater than 887? 0.3 % (Round to one decimal place as needed.) (c) What percentage of standardized test scores is greater than 768? % (Round to one decimal place as needed.)

Solution Steps

To find the percentage of standardized test scores between \(173\) and \(887\), we calculate the probability \(P\) as follows:

\[ P = \Phi(Z_{end}) - \Phi(Z_{start}) = \Phi(3.0) - \Phi(-3.0) = 1.0 \]

Thus, the percentage of scores between \(173\) and \(887\) is:

\[ \text{Percentage} = P \times 100 = 100.0\% \]

The percentage of scores less than \(173\) or greater than \(887\) is the complement of the percentage calculated in Step 1:

\[ \text{Percentage} = 100\% - 100.0\% = 0.0\% \]

To find the percentage of scores greater than \(768\), we again calculate the probability \(P\):

\[ P = \Phi(Z_{end}) - \Phi(Z_{start}) = \Phi(\infty) - \Phi(2.0) = 0.0 \]

Thus, the percentage of scores greater than \(768\) is:

\[ \text{Percentage} = (1 - P) \times 100 = 100.0\% \]

Final Answer