Step-by-Step Solution

Part 1: Formulate the Hypotheses

The hypotheses for this test are as follows:

• Null hypothesis ($H_0$): $\sigma = 0.003$ (The standard deviation has not decreased.)
• Alternative hypothesis ($H_1$): $\sigma < 0.003$ (The standard deviation has decreased.)

These hypotheses represent a left-tailed test since we are testing for a decrease in the standard deviation.

Part 2: Test Statistic

The test statistic for the Chi-Square test of standard deviation is calculated using the formula:

$\chi^2 = \frac{(n - 1) \cdot s^2}{\sigma_0^2}$

Where:

• $n = 22$ (sample size)
• $s = 0.0026$ (sample standard deviation)
• $\sigma_0 = 0.003$ (population standard deviation under $H_0$)

Substitute the values:

$\chi^2 = \frac{(22 - 1) \cdot (0.0026)^2}{(0.003)^2}$

$\chi^2 = \frac{21 \cdot 0.00000676}{0.000009}$

$\chi^2 \approx 15.773$

Thus, the value of the test statistic is:

$\chi^2 = 15.773$

Part 3: $P$-value Calculation

To find the $P$-value, we use the Chi-Square distribution with $df = n - 1 = 21$. The $P$-value corresponds to the left-tailed probability for $\chi^2 = 15.773$.

Using technology or statistical tables, we find:

$P\text{-value} \approx 0.218$

Part 4: Conclusion

At the $\alpha = 0.01$ level of significance:

• The $P$-value ($0.218$) is greater than the significance level ($0.01$).
• Therefore, we do not reject the null hypothesis.

Conclusion: There is not sufficient evidence to conclude that the standard deviation has decreased at the $\alpha = 0.01$ level of significance.

Summary of Results:

1. Hypotheses:

   • $H_0: \sigma = 0.003$
   • $H_1: \sigma < 0.003$

2. Test statistic: $\chi^2 = 15.773$

3. $P$-value: $0.218$

4. Conclusion: Do not reject $H_0$. No significant evidence to conclude a decrease in the standard deviation.

Questions for Further Understanding:

1. Why is a Chi-Square test appropriate for testing variance or standard deviation?
2. What does the $P$-value represent in the context of this test?
3. How would the conclusion change if the significance level were $0.05$ instead of $0.01$?
4. What are the assumptions of the Chi-Square test, and are they satisfied in this scenario?
5. How would the analysis change if the alternative hypothesis were $\sigma \neq 0.003$ (two-tailed test)?

Tip:

When interpreting $P$-values, always compare them to the significance level ($\alpha$) carefully to make an accurate decision about the null hypothesis.