If the sample standard deviation is 2.77, what would be the estimated standard error of sample means for sample size 15? Write the result with 2 decimal places.
The correct answer and explanation is:
To calculate the estimated standard error of the sample mean, we use the following formula: Standard Error (SE)=sn\text{Standard Error (SE)} = \frac{s}{\sqrt{n}}
Where:
- ss is the sample standard deviation
- nn is the sample size
Given:
- s=2.77s = 2.77
- n=15n = 15
Now compute the square root of the sample size: 15≈3.873\sqrt{15} \approx 3.873
Then divide the sample standard deviation by the square root of the sample size: SE=2.773.873≈0.7152SE = \frac{2.77}{3.873} \approx 0.7152
Rounded to two decimal places, the estimated standard error is: 0.72\boxed{0.72}
📘 Explanation (300 Words):
The standard error of the mean (SE) is a measure that tells us how much variability we can expect in sample means if we repeatedly took samples of the same size from a population. In other words, it reflects the precision of the sample mean as an estimate of the population mean.
When we take a sample from a population, the sample standard deviation (s) measures the spread or variability within that sample. However, to assess the accuracy of the sample mean, we need to understand how the mean might fluctuate from one sample to another — that’s where the standard error comes in.
Mathematically, the standard error decreases as the sample size increases. This is because a larger sample provides more information and reduces the impact of random variation.
In this case, you have a sample of size 15 and a sample standard deviation of 2.77. Plugging those values into the formula for standard error, we divide 2.77 by the square root of 15 (approximately 3.873), giving us a standard error of approximately 0.7152. Rounding to two decimal places, we get 0.72.
This value tells us that if we were to repeatedly draw samples of size 15 from the same population, the standard deviation of the sample means would be about 0.72. This gives insight into how reliable your sample mean is as an estimate of the population mean.