Answer :
Using the 68-95-99.7 rule for a normal distribution with a mean of 40 and a standard deviation of 3, the percentage of values between 34 and 40 is 47.5%.
According to the 68-95-99.7 rule (also known as the Empirical Rule), for a normal distribution, 68% of the data falls within one standard deviation of the mean. This means that 34% of the data lies between the mean and one standard deviation below the mean, and another 34% lies between the mean and one standard deviation above the mean. In this case, the mean is 40 and the standard deviation is 3. Therefore, 34% of the values lie between 37 (one standard deviation below the mean) and 40 (the mean itself).
However, we are interested in the percentage of values between 34 and 40. Thirty-four is two standard deviations below the mean. According to the 68-95-99.7 rule, 95% of data lies within two standard deviations of the mean, which means approximately 47.5% lie below the mean. Subtracting the 34% that lies between one standard deviation below and the mean itself, we get 47.5% - 34% = 13.5%. This represents the percentage of values lying between two standard deviations and one standard deviation below the mean.
Therefore, the percentage of values in the distribution between 34 and 40 is 34% (between mean and one standard deviation) plus 13.5% (between two standard deviations and one standard deviation below the mean), which equals 47.5%.
