Find the 90 confidence interval for the variance and standard deviation for the lifetime of
disposable camera batteries if a sample of 19 disposable camera batteries has a standard
deviation of 2 1 months Assume the variable is normally distributed Do you feel that the
lifetimes of the batteries are relatively consistent?

To find the 90% confidence interval for the variance and standard deviation of the lifetime of disposable camera batteries, given that the sample standard deviation \( s \) is 2.1 months and the sample size \( n \) is 19, we need to use the chi-square distribution. The steps are as follows:

1. **Calculate the sample variance:**
   \[
   s^2 = (2.1)^2 = 4.41 \text{ months}^2
   \]

2. **Determine the degrees of freedom:**
   \[
   df = n - 1 = 19 - 1 = 18
   \]

3. **Find the chi-square critical values for the 90% confidence interval:**
   For a 90% confidence interval with 18 degrees of freedom, we need the chi-square values for the lower 5% and upper 5%. These values can be found in chi-square distribution tables or using statistical software.

   \[
   \chi^2_{\alpha/2, \, df} = \chi^2_{0.05, \, 18} \approx 10.117
   \]
   \[
   \chi^2_{1-\alpha/2, \, df} = \chi^2_{0.95, \, 18} \approx 28.869
   \]

4. **Calculate the confidence interval for the variance:**
   The confidence interval for the variance \(\sigma^2\) is given by:
   \[
   \left( \frac{(n-1)s^2}{\chi^2_{1-\alpha/2}}, \frac{(n-1)s^2}{\chi^2_{\alpha/2}} \right)
   \]
   Substituting the values:
   \[
   \left( \frac{18 \cdot 4.41}{28.869}, \frac{18 \cdot 4.41}{10.117} \right)
   \]
   \[
   \left( \frac{79.38}{28.869}, \frac{79.38}{10.117} \right)
   \]
   \[
   \left( 2.75, 7.84 \right)
   \]

5. **Calculate the confidence interval for the standard deviation:**
   The confidence interval for the standard deviation \(\sigma\) is the square root of the variance interval:
   \[
   \left( \sqrt{2.75}, \sqrt{7.84} \right)
   \]
   \[
   \left( 1.66, 2.80 \right)
   \]

Therefore, the 90% confidence interval for the variance is \((2.75, 7.84) \text{ months}^2\) and for the standard deviation is \((1.66, 2.80) \text{ months}\).

### Consistency of Battery Lifetimes
To determine if the lifetimes of the batteries are relatively consistent, we look at the width of the confidence interval for the standard deviation. A narrower interval indicates more consistency. In this case, the interval \((1.66, 2.80)\) suggests that while there is some variation in battery lifetimes, it is not excessively wide, indicating a moderate level of consistency.

Would you like more details or have any questions? Here are some further questions you might want to explore:

1. How do you interpret the confidence interval for the standard deviation?
2. What assumptions are required for the chi-square confidence interval to be valid?
3. How would you calculate the confidence interval if the sample size were larger?
4. How does the confidence level affect the width of the confidence interval?
5. What are some potential sources of variability in the lifetime of disposable camera batteries?

**Tip:** Always verify the assumptions of normality when using statistical methods that require it, as violating these assumptions can affect the accuracy of your results.

Learn how to calculate the 90% confidence interval for the variance and standard deviation of disposable camera battery lifetimes. Understand the statistical methods involving sample variance, chi-square distribution, and interpretation of confidence intervals.

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