The following are weights (in lbs) for a sample of 11 baby boys at the age of 18 months:
21.8
22.2
23.4
24.6
25.1
26.7
27.3
28.0
29.8
30.9
31.2
What is the 80th percentile estimate for these sample?
Find the mean first = sum of scores/number of scores
Subtract each of the scores from the mean and square each difference. Find the sum of these squares. Divide that by the number of scores to get variance.
Standard deviation = square root of variance
Z = (score-mean)/SD
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability (.80) and its Z score. Insert Z score into equation above.
To find the 80th percentile estimate for the sample, you need to arrange the weights in ascending order:
21.8, 22.2, 23.4, 24.6, 25.1, 26.7, 27.3, 28.0, 29.8, 30.9, 31.2
Since the sample size is 11, the 80th percentile corresponds to the value at position (11+1) * 0.8 = 9.6, which rounds up to position 10.
The 10th value in the ordered list is 30.9 lbs. Hence, the 80th percentile estimate for this sample is 30.9 lbs.
To find the 80th percentile estimate for the sample weights, you need to sort the data from smallest to largest. Here is the sorted list:
21.8, 22.2, 23.4, 24.6, 25.1, 26.7, 27.3, 28.0, 29.8, 30.9, 31.2
Next, you need to calculate the rank of the percentile you're interested in. In this case, since you want the 80th percentile, you need to find the rank that corresponds to 80%. To do this, you can use the formula:
Rank = (p / 100) * (n + 1)
where p represents the desired percentile (80 in this case), and n represents the number of observations in the sample (11 in this case).
Rank = (80 / 100) * (11 + 1) = 0.8 * 12 = 9.6
Since the rank is not a whole number, you need to use interpolation to estimate the 80th percentile. Interpolation involves finding the two nearest ranks around the desired percentile and calculating a weighted average.
The nearest ranks around rank 9.6 are ranks 9 and 10. The weight for rank 9.6 is calculated as the difference between the desired rank and the lower rank divided by the difference between the higher rank and the lower rank:
Weight = (9.6 - 9) / (10 - 9) = 0.6
Now, you can estimate the 80th percentile by using the weights and corresponding values of the nearest ranks:
80th percentile estimate = (weight * value at rank 9) + ((1 - weight) * value at rank 10)
80th percentile estimate = (0.6 * 29.8) + (0.4 * 30.9) = 17.88 + 12.36 = 30.24
So, the estimated 80th percentile for the sample of baby boy weights at 18 months old is approximately 30.24 lbs.