To calculate the 99% confidence interval for the number of chocolate chips per cookie, we will use the formula for the confidence interval when the population standard deviation is unknown, which requires the t-distribution:
\[ \text{Confidence Interval} = \bar{x} \pm t^* \left(\frac{s}{\sqrt{n}}\right) \]
Where:
- \(\bar{x} = 6.3\) (sample mean)
- \(s = 3.1\) (sample standard deviation)
- \(n = 46\) (sample size)
- \(t^*\) is the critical t-value for a 99% confidence level, with degrees of freedom \(df = n - 1 = 46 - 1 = 45\).
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Finding the critical t-value (\(t^*\)): To find the critical t-value for a 99% confidence level and 45 degrees of freedom, we can refer to a t-table or use statistical software/calculator. The critical t-value for 99% confidence level (two-tailed) is approximately:
\(t^* \approx 2.680\)
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Calculating the standard error (SE): \[ SE = \frac{s}{\sqrt{n}} = \frac{3.1}{\sqrt{46}} \approx \frac{3.1}{6.782} \approx 0.456 \]
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Calculating the margin of error (ME): \[ ME = t^* \times SE = 2.680 \times 0.456 \approx 1.224 \]
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Calculating the confidence interval:
- Lower limit: \[ \bar{x} - ME = 6.3 - 1.224 \approx 5.1 \]
- Upper limit: \[ \bar{x} + ME = 6.3 + 1.224 \approx 7.5 \]
Therefore, the 99% confidence interval for the number of chocolate chips per cookie for Big Chip cookies is:
\[ (5.1, 7.5) \]
In summary, rounding to one decimal place:
The 99% confidence interval is (5.1, 7.5).