To calculate the margin of error (M.E.) for estimating a population mean using a sample, you can use the formula:
\[ \text{M.E.} = z \times \left( \frac{s}{\sqrt{n}} \right) \]
where:
- \( z \) is the z-score corresponding to the desired confidence level,
- \( s \) is the sample standard deviation,
- \( n \) is the sample size.
Step 1: Identify the z-score
For a 90% confidence level, the z-score can be found using a standard normal distribution table or calculator. The critical value for a 90% confidence level (which corresponds to 5% in each tail) is:
\[ z \approx 1.645 \]
Step 2: Calculate the standard error
The standard error (SE) is calculated using the formula:
\[ SE = \frac{s}{\sqrt{n}} \]
Substituting the given values:
- \( s = 12.6 \)
- \( n = 477 \)
Calculating \( SE \):
\[ SE = \frac{12.6}{\sqrt{477}} \approx \frac{12.6}{21.82} \approx 0.577 \]
Step 3: Calculate the Margin of Error
Now substituting \( z \) and \( SE \) into the margin of error formula:
\[ \text{M.E.} = 1.645 \times 0.577 \approx 0.950 \]
Step 4: Round the Margin of Error
Rounding to one decimal place:
\[ \text{M.E.} \approx 1.0 \]
Therefore, the margin of error at a confidence level of 90% is:
\[ \text{M.E.} = 1.0 \]
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