A battery manufacturer is positive that its AA batteries last for 1,000 hours. However, you’re working for Consumer Reports and you don’t agree with what they say about their product. So, you buy 400 batteries and you test them out. Your findings are that the batteries last 990 hours on average with a standard deviation of 125. If you used a significance level of 1%, what is your opinion about the company’s batteries?

To determine your opinion about the company's batteries, you can conduct a hypothesis test. The null hypothesis (H0) in this case would be that the batteries last for 1,000 hours, while the alternative hypothesis (H1) would be that the batteries do not last for 1,000 hours.

Here's how you can calculate the hypothesis test and draw a conclusion:

1. Formulate the null and alternative hypotheses:
- Null hypothesis (H0): The average battery life is equal to 1,000 hours.
- Alternative hypothesis (H1): The average battery life is not equal to 1,000 hours.

2. Select the appropriate test statistic:
Since you have a sample mean and a known standard deviation, you can use the z-test statistic.

3. Determine the significance level:
The significance level, also known as alpha (α), is given in the problem as 1%. This means you want your test to have a 99% confidence level.

4. Calculate the test statistic:
Using the formula: z = (sample mean - population mean) / (standard deviation / √sample size)

z = (990 - 1000) / (125 / √400)
z = -10 / (125 / 20)
z = -10 / 6.25
z = -1.6

5. Determine the critical value:
With a significance level of 1%, the critical values (critical regions) for a two-tailed test can be found using a standard normal distribution table or a z-table. For a 99% confidence level, it corresponds to a critical value of ±2.576.

6. Compare the test statistic with the critical value:
Since -1.6 falls within the range -2.576 to 2.576, you do not reject the null hypothesis.

7. Draw a conclusion:
Based on the statistical analysis, you do not have sufficient evidence to conclude that the average battery life is significantly different from 1,000 hours. Therefore, you cannot reject the company's claim that the AA batteries last for 1,000 hours.

In summary, your opinion about the company's batteries is that they do indeed last for approximately 1,000 hours based on the statistical analysis conducted.