The problem is asking for the null hypothesis ($H_0$) and the alternative hypothesis ($H_1$) for a hypothesis test regarding the standard deviation of the lifetimes of electric bulbs produced by BIG Corporation. Let's break this down:

1. Given Information:

   • Published claim: The standard deviation ($\sigma$) of the lifetimes is 121.5.
   • BIG Corporation's claim: The standard deviation is less than 121.5.

2. Hypotheses:

   • Null Hypothesis ($H_0$): The standard deviation is equal to 121.5.
     • $H_0: \sigma = 121.5$
   • Alternative Hypothesis ($H_1$): The standard deviation is less than 121.5.
     • $H_1: \sigma < 121.5$

This is a one-tailed hypothesis test because the alternative hypothesis specifies a direction (less than).

Would you like further explanation of hypothesis testing, or how this would be tested using a chi-square test for variance? Here are some additional questions to deepen understanding:

1. How is the chi-square distribution used to test hypotheses about variance or standard deviation?
2. What is the significance level ($\alpha$) typically chosen for such tests?
3. How does one calculate the test statistic for standard deviation hypothesis tests?
4. Why is the null hypothesis always phrased as an equality?
5. How would the critical value for the test be determined?

Tip: Always remember to check whether the data meets the assumptions of the test (e.g., normality) before proceeding with hypothesis testing.