Question 1177447
We're going to conduct a hypothesis test for a mean.

Let $\mu$ be the true mean volume of cola in the bottles.

1. **Hypotheses:**
   * Null hypothesis: $\mu = 32$ (The company is not cheating.)
   * Alternative hypothesis: $\mu < 32$ (The company is cheating.)

2. **Test statistic:**
   We use the t-statistic, since the population standard deviation is unknown:
   ```
   t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}}
   ```
   where $\bar{x} = 31.4$ is the sample mean, $\mu_0 = 32$ is the hypothesized mean, $s = 1.75$ is the sample standard deviation, and $n = 50$ is the sample size.  This gives us
   ```
   t = \frac{31.4 - 32}{1.75 / \sqrt{50}} \approx -2.42.
   ```

3. **P-value:**
   The p-value is the probability of observing a sample mean as extreme as $\bar{x} = 31.4$, assuming the null hypothesis is true.  Since this is a left-tailed test, the p-value is
   ```
   P(T < -2.42) \approx 0.0096,
   ```
   where $T$ has a t-distribution with $n - 1 = 49$ degrees of freedom.

4. **Conclusion:**
   Since the p-value (0.0096) is less than $\alpha = 0.01$, we reject the null hypothesis.  There is enough evidence to conclude that the company is cheating the consumers.