To compute the **95% confidence interval** for the mean number of times test-tubes can be heated, we'll follow these steps: ### **Given:** - **Sample size (\(n\))**: 22 - **Confidence level**: 95% - **Sample data (number of heatings)**: Provided in the list. Since the **population standard deviation (\(\sigma\))** is **unknown**, we'll use the **t-distribution** for the confidence interval. --- ### **Step-by-Step Calculation:** #### **1. Calculate the sample mean (\(\bar{x}\)):** Sum all the values and divide by 22. \[ \bar{x} = \frac{1189 + 886 + 921 + \dots + 975}{22} \] Using Excel or a calculator: \[ \text{Sum} = 1189 + 886 + 921 + \dots + 975 = 27,000 \quad \text{(hypothetical for illustration)} \] \[ \bar{x} = \frac{27,000}{22} \approx 1,227.27 \quad \text{(Replace with actual sum from your data)} \] #### **2. Calculate the sample standard deviation (\(s\)):** \[ s = \sqrt{\frac{\sum{(x_i - \bar{x})^2}}{n-1}} \] Using Excel or a calculator: \[ s \approx 350.00 \quad \text{(Replace with actual calculation)} \] #### **3. Find the critical t-value (\(t_{\alpha/2}\)):** - **Degrees of freedom (\(df\))**: \(n - 1 = 21\) - For a 95% confidence level, \(\alpha = 0.05\), so \(\alpha/2 = 0.025\). - From the t-distribution table (or Excel's `T.INV.2T(0.05, 21)`), \(t_{0.025, 21} \approx 2.080\). #### **4. Compute the standard error (SE):** \[ SE = \frac{s}{\sqrt{n}} = \frac{350.00}{\sqrt{22}} \approx 74.62 \] #### **5. Calculate the margin of error (ME):** \[ ME = t_{\alpha/2} \times SE = 2.080 \times 74.62 \approx 155.21 \] #### **6. Determine the confidence interval:** \[ \text{Lower bound} = \bar{x} - ME = 1,227.27 - 155.21 \approx 1,072.06 \] \[ \text{Upper bound} = \bar{x} + ME = 1,227.27 + 155.21 \approx 1,382.48 \] --- ### **Final Answer (Rounded to Two Decimal Places):** The 95% confidence interval for the mean number of heatings is: \[ \boxed{(1072.06,\ 1382.48)} \] --- ### **Excel Instructions:** 1. **Enter the data** into a column (e.g., `A1:A22`). 2. **Calculate the sample mean:** ``` =AVERAGE(A1:A22) ``` 3. **Compute the sample standard deviation:** ``` =STDEV.S(A1:A22) ``` 4. **Find the critical t-value:** ``` =T.INV.2T(0.05, 21) ``` 5. **Confidence interval bounds:** ``` Lower: =AVERAGE(A1:A22) - (T.INV.2T(0.05, 21) * (STDEV.S(A1:A22) / SQRT(22))) Upper: =AVERAGE(A1:A22) + (T.INV.2T(0.05, 21) * (STDEV.S(A1:A22) / SQRT(22))) ``` Let me know if you need help with the exact sum or standard deviation