Assume that a set of test scores is normally distributed with a mean of 120 and a standard deviation of 25. Use the $68-95-99.7$ rule to find the following quantities.
a. The percentage of scores less than 120 is $\square \%$. (Round to one decimal place as needed.)
b. The percentage of scores greater than 145 is $\%$.
(Round to one decimal place as needed.)
c. The percentage of scores between 70 and 145 is $\%$ (Round to one decimal place as needed.)
Answer
Final Answer: a. The percentage of scores less than 120 is \(\boxed{50\%}\). b. The percentage of scores greater than 145 is \(\boxed{16\%}\). c. The percentage of scores between 70 and 145 is \(\boxed{27\%}\).
Step 1 ：Assume that a set of test scores is normally distributed with a mean of 120 and a standard deviation of 25. Use the $68-95-99.7$ rule to find the following quantities.
Step 2 ：The $68-95-99.7$ rule, also known as the empirical rule, states that for a normal distribution, almost all values lie within 3 standard deviations of the mean. More specifically, 68% of the data falls within the first standard deviation from the mean, 95% fall within two standard deviations, and 99.7% fall within three standard deviations.
Step 3 ：The percentage of scores less than 120 is 50%. This is because 120 is the mean of the distribution, and in a normal distribution, 50% of the data is less than the mean and 50% is greater than the mean.
Step 4 ：The percentage of scores greater than 145 can be found by first determining how many standard deviations away 145 is from the mean. 145 is one standard deviation away from the mean (since the standard deviation is 25), so according to the empirical rule, 68% of the data falls within one standard deviation of the mean. This means that 32% of the data falls outside this range. Since a normal distribution is symmetric, half of this 32% is greater than one standard deviation above the mean and half is less than one standard deviation below the mean. So, the percentage of scores greater than 145 is 16%.
Step 5 ：The percentage of scores between 70 and 145 can be found by determining how many standard deviations away 70 and 145 are from the mean. 70 is two standard deviations below the mean and 145 is one standard deviation above the mean. According to the empirical rule, 95% of the data falls within two standard deviations of the mean and 68% falls within one standard deviation. So, the percentage of scores between 70 and 145 is the difference between these two percentages, which is 27%.
Step 6 ：Final Answer: a. The percentage of scores less than 120 is \(\boxed{50\%}\). b. The percentage of scores greater than 145 is \(\boxed{16\%}\). c. The percentage of scores between 70 and 145 is \(\boxed{27\%}\).
