To calculate a 90% confidence interval for the mean mass of dog food, you typically need the following information:
- Sample Mean (\( \bar{x} \)): The average mass from a sample of dog food.
- Sample Standard Deviation (\( s \)): The standard deviation of the sample masses.
- Sample Size (\( n \)): The number of observations in the sample.
- Critical Value (\( t^ \))*: The t-score that corresponds to the desired confidence level (for a 90% confidence interval, you can find this from the t-distribution table based on the degrees of freedom, which is \( df = n - 1 \)).
Here are the steps to find the confidence interval:
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Determine the critical t-value. For 90% confidence and \( df = n - 1 \), you can usually find this in a t-table or through statistical software.
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Calculate the Standard Error (SE): \[ SE = \frac{s}{\sqrt{n}} \]
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Calculate the confidence interval: \[ CI = \bar{x} \pm t^* \cdot SE \]
Example Calculation
Let's say you have the following values:
- Sample Mean (\( \bar{x} \)) = 50 grams
- Sample Standard Deviation (\( s \)) = 5 grams
- Sample Size (\( n \)) = 20
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Find the critical t-value: For a 90% confidence interval with \( n - 1 = 19 \) degrees of freedom, the t-value (approximately) is 1.645 (you can confirm this with a t-table or software).
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Calculate the Standard Error (SE): \[ SE = \frac{s}{\sqrt{n}} = \frac{5}{\sqrt{20}} \approx 1.118 \]
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Calculate the confidence interval: \[ CI = 50 \pm 1.645 \cdot 1.118 \] \[ CI = 50 \pm 1.839 \] \[ CI = (48.161, 51.839) \]
Thus, the 90% confidence interval for the mean mass of all dog food in this case would be approximately \( (48.16, 51.84) \) grams.
Make sure to replace the example values with your actual sample values for your specific calculation!