Final answer:
To create the set, select six positive integers with a sum of 60 to achieve a mean of 10, ensure the difference between the third and first quartiles is 14 for an IQR of 14, and calculate to find a population standard deviation between 7 and 8, such as the set {1, 3, 9, 11, 17, 19}.
Step-by-step explanation:
To create a set of 6 positive integers whose mean is 10, whose IQR is 14, and whose population standard deviation is between 7 and 8, we will need to ensure that our set not only satisfies the mean requirement but also the spread of the data as indicated by the IQR and standard deviation. Let's denote our set of integers as {a, b, c, d, e, f} ordered from smallest to largest.
The mean is the sum of all values divided by the number of values, so we need the sum to be 60 (6 times 10). To achieve an IQR of 14, we want the difference between the third quartile (75th percentile, which will be e) and the first quartile (25th percentile, which will be b) to be 14.
An example set fulfilling these conditions could be {1, 3, 9, 11, 17, 19}. This set has a mean of 10 ((1+3+9+11+17+19)/6 = 60/6 = 10), an IQR of 14 (17-3 = 14), and a population standard deviation of approximately 7.48, which is within the range of 7 to 8.
To check the standard deviation, we use the formula for the population standard deviation which accounts for each value's deviation from the mean, and since there is a small number of data points, we can calculate this manually or with a calculator to ensure it falls within the specified range.