Question 1195251: Manufacturer of pacemaker wants the standard deviation of the life time of the batteries to be less than 1.8 months. a sample of 20 batteries had standard deviation of 1.6 months. assume the variable is normally distributed. find the 95% confidence interval of standard deviation of batteries. based on this answer do you feel that 1.8 months is reasonable estimate?
Answer by proyaop(69)   (Show Source):
You can put this solution on YOUR website! **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.
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