To calculate the 95% confidence interval for the number of chocolate chips per cookie, we can use the formula for the confidence interval of the mean:
\[ \text{Confidence Interval} = \bar{x} \pm z \left( \frac{s}{\sqrt{n}} \right) \]
where:
- \(\bar{x}\) is the sample mean,
- \(z\) is the z-value for the desired confidence level,
- \(s\) is the sample standard deviation,
- \(n\) is the sample size.
From the problem:
- \(\bar{x} = 8.9\)
- \(s = 3.5\)
- \(n = 54\)
Step 1: Find the z-value for a 95% confidence level
For a 95% confidence level, the z-value (the critical value) from the standard normal distribution is approximately 1.96.
Step 2: Calculate the standard error (SE)
The standard error (SE) is calculated as follows:
\[ SE = \frac{s}{\sqrt{n}} = \frac{3.5}{\sqrt{54}} \approx \frac{3.5}{7.348} \approx 0.476 \]
Step 3: Calculate the margin of error (ME)
Now, calculate the margin of error:
\[ ME = z \cdot SE = 1.96 \cdot 0.476 \approx 0.93456 \]
Step 4: Calculate the confidence interval
Using the margin of error, the confidence interval is:
\[ \text{Confidence Interval} = \bar{x} \pm ME = 8.9 \pm 0.93456 \]
Calculating the lower and upper bounds:
\[ \text{Lower Bound} = 8.9 - 0.93456 \approx 7.96544 \quad \text{(approximately 8.0)} \]
\[ \text{Upper Bound} = 8.9 + 0.93456 \approx 9.83456 \quad \text{(approximately 9.8)} \]
Conclusion
Thus, the 95% confidence interval for the number of chocolate chips per cookie is:
\[ \boxed{(8.0, 9.8)} \]
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