Question 1195251
**1. Find the Chi-Square Values**

* **Degrees of Freedom (df):** df = n - 1 = 20 - 1 = 19
* **Confidence Level:** 95% 
* **Significance Level (α):** α = 1 - 0.95 = 0.05
* **α/2:** 0.05 / 2 = 0.025

* **Find Chi-Square Values:**
    * **Chi-Square Lower (χ²lower):** Look up the value in a Chi-Square distribution table for df = 19 and α/2 = 0.025. 
        * χ²lower ≈ 8.907 
    * **Chi-Square Upper (χ²upper):** Look up the value in a Chi-Square distribution table for df = 19 and 1 - α/2 = 0.975.
        * χ²upper ≈ 32.852

**2. Calculate the Confidence Interval for the Variance**

* **Lower Limit of Variance:** (n - 1) * s² / χ²upper 
    * = (20 - 1) * 1.6² / 32.852 
    * ≈ 1.47 
* **Upper Limit of Variance:** (n - 1) * s² / χ²lower
    * = (20 - 1) * 1.6² / 8.907 
    * ≈ 5.48

**3. Calculate the Confidence Interval for the Standard Deviation**

* **Lower Limit of Standard Deviation:** √(Lower Limit of Variance) 
    * = √1.47 
    * ≈ 1.21 months
* **Upper Limit of Standard Deviation:** √(Upper Limit of Variance) 
    * = √5.48 
    * ≈ 2.34 months

**4. Interpretation**

The 95% confidence interval for the standard deviation of battery lifetimes is approximately **(1.21 months, 2.34 months)**.

**5. Evaluation of the Manufacturer's Claim**

The manufacturer desires a standard deviation less than 1.8 months. 

* **Observation:** The upper limit of the confidence interval (2.34 months) is greater than 1.8 months.

* **Conclusion:** Based on this sample and the 95% confidence interval, we cannot conclusively say that the standard deviation of battery lifetimes is less than 1.8 months. There is a possibility that the true population standard deviation is higher. 

**Further Considerations:**

* This analysis assumes that the sample of batteries is truly representative of the entire population of batteries produced by the manufacturer. 
* If the manufacturer's target is critical for safety or reliability, further investigation and potentially a larger sample size may be necessary to make a more definitive conclusion.

**Disclaimer:** 

This analysis provides a statistical interpretation. The manufacturer should consider this information alongside other factors (e.g., cost, customer expectations) when making decisions about battery production and quality control.