To construct a 95% confidence interval for the mean percentage of fake followers, we can use the formula for the confidence interval for the mean when the standard deviation is known. Given the sample size, mean, and standard deviation, we will follow these steps:
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Identify the sample values:
- Sample size (\( n \)) = 44
- Sample mean (\( \bar{x} \)) = 14.3%
- Sample standard deviation (\( s \)) = 10.3%
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Determine the critical value: Since the sample size is 44, we can use the t-distribution to find the critical value. For a 95% confidence interval and degrees of freedom \( n-1 = 44-1 = 43 \), we can look up the t-value or use statistical software/calculator.
The two-tailed critical t-value for \( df = 43 \) at 95% confidence level is approximately \( t_{0.025}(43) \approx 2.0167 \) (exact values can be looked up in a t-distribution table or using software).
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Calculate the standard error (SE): \[ SE = \frac{s}{\sqrt{n}} = \frac{10.3}{\sqrt{44}} \approx \frac{10.3}{6.633} \approx 1.55 \]
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Construct the confidence interval: The formula for the confidence interval is: \[ \bar{x} \pm t_{critical} \times SE \] Plugging in the values: \[ CI = 14.3 \pm 2.0167 \times 1.55 \] \[ CI = 14.3 \pm 3.125 \]
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Calculate the lower and upper bounds: \[ \text{Lower bound} = 14.3 - 3.125 \approx 11.175 \] \[ \text{Upper bound} = 14.3 + 3.125 \approx 17.425 \]
Thus, the 95% confidence interval for the mean percentage of fake followers is approximately:
\[ \boxed{(11.18%, 17.43%)} \]
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