Final answer:
To determine which ages fall within two standard deviations of the mean, we need to calculate the mean and standard deviation of the given ages. The mean is 61.625 and the standard deviation is 10.502. Ages that fall within two standard deviations are: 42, 48, 54, 54, 59, 60, 65, 68, 77, 83, and 88.
Step-by-step explanation:
To determine which ages fall within two standard deviations, we need to calculate the mean and standard deviation of the given ages. The mean is the sum of all the ages divided by the number of ages, which is 16 in this case. The standard deviation measures the spread of the ages around the mean. Once we have the mean and standard deviation, we can identify the ages that fall within two standard deviations from the mean.
Let's calculate the mean first:
Mean = (34 + 18 + 28 + 54 + 60 + 53 + 48 + 88 + 65 + 77 + 59 + 68 + 54 + 42 + 83 + 90) / 16 = 61.625
Now, let's calculate the standard deviation:
Step 1: Calculate the difference between each age and the mean:
(34 - 61.625), (18 - 61.625), (28 - 61.625), (54 - 61.625), (60 - 61.625), (53 - 61.625), (48 - 61.625), (88 - 61.625), (65 - 61.625), (77 - 61.625), (59 - 61.625), (68 - 61.625), (54 - 61.625), (42 - 61.625), (83 - 61.625), (90 - 61.625)
Step 2: Square each difference:
(-27.625)^2, (-43.625)^2, (-33.625)^2, (-7.625)^2, (-1.625)^2, (-8.625)^2, (-13.625)^2, (26.375)^2, (3.375)^2, (15.375)^2, (-2.625)^2, (6.375)^2, (-7.625)^2, (-19.625)^2, (21.375)^2, (28.375)^2
Step 3: Calculate the sum of the squared differences:
(-27.625)^2 + (-43.625)^2 + (-33.625)^2 + (-7.625)^2 + (-1.625)^2 + (-8.625)^2 + (-13.625)^2 + (26.375)^2 + (3.375)^2 + (15.375)^2 + (-2.625)^2 + (6.375)^2 + (-7.625)^2 + (-19.625)^2 + (21.375)^2 + (28.375)^2 = 6965
Step 4: Divide the sum of squared differences by the number of ages minus one, then take the square root of the result:
Square root of (6965 / (16 - 1)) = 10.502
Finally, we can identify the ages that fall within two standard deviations from the mean. Two standard deviations above the mean would be 61.625 + 2 * 10.502 = 82.629, and two standard deviations below the mean would be 61.625 - 2 * 10.502 = 40.621. Therefore, the ages that fall within two standard deviations are: 42, 48, 54, 54, 59, 60, 65, 68, 77, 83, and 88.