As the sample size increases, the standard error of the mean (M) decreases. This is because the standard error is inversely proportional to the square root of the sample size. To demonstrate this, we can calculate the standard error for different sample sizes using Python.
The standard error (SE) of the mean M M M is calculated using the formula: SE=σn SE = \frac{\sigma}{\sqrt{n}} SE=nσ where σ \sigma σ is the standard deviation and n n n is the sample size. As n n n increases, SE SE SE decreases.
Using a standard deviation σ=10 \sigma = 10 σ=10, we calculate the standard error for various sample sizes:
For n=10 n = 10 n=10: SE=1010≈3.1623 SE = \frac{10}{\sqrt{10}} \approx 3.1623 SE=1010≈3.1623
For n=50 n = 50 n=50: SE=1050≈1.4142 SE = \frac{10}{\sqrt{50}} \approx 1.4142 SE=5010≈1.4142
For n=100 n = 100 n=100: SE=10100=1.00 SE = \frac{10}{\sqrt{100}} = 1.00 SE=10010=1.00
For n=500 n = 500 n=500: SE=10500≈0.4472 SE = \frac{10}{\sqrt{500}} \approx 0.4472 SE=50010≈0.4472
For n=1000 n = 1000 n=1000: SE=101000≈0.3162 SE = \frac{10}{\sqrt{1000}} \approx 0.3162 SE=100010≈0.3162
The calculated standard errors for the different sample sizes are as follows:
The standard error decreases as the sample size increases, with the following values:
Thus, the final answer is: SE values: 3.16,1.41,1.00 \boxed{SE \text{ values: } 3.16, 1.41, 1.00} SE values: 3.16,1.41,1.00
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