A random sample of 

6464 

cans of soda is obtained and the contents are measured. The sample mean is 

12.0312.03 

oz and the standard deviation is 

0.120.12 

oz. Test the claim that the contents of all such cans have a mean different from 

12.0012.00 

​oz, as indicated by the label. Use a 0.05 significance level.

To test the claim that the mean content of all such cans differs from 12.00 oz, we'll conduct a hypothesis test using the following steps:

### 1. **State the Hypotheses**

- **Null Hypothesis (\(H_0\))**: The mean content of all cans is 12.00 oz.  
  \[
  H_0: \mu = 12.00
  \]

- **Alternative Hypothesis (\(H_1\))**: The mean content of all cans is different from 12.00 oz.  
  \[
  H_1: \mu \neq 12.00
  \]

This is a two-tailed test because we are testing for a mean different from 12.00 oz, not just greater than or less than.

### 2. **Determine the Test Statistic**

Given:
- Sample mean (\(\bar{x}\)) = 12.03 oz
- Population mean under the null hypothesis (\(\mu_0\)) = 12.00 oz
- Sample standard deviation (\(s\)) = 0.12 oz
- Sample size (\(n\)) = 64

We'll use the **t-test** for the mean, but since the sample size is large (\(n > 30\)), the **z-test** is appropriate:

\[
z = \frac{\bar{x} - \mu_0}{\frac{s}{\sqrt{n}}}
\]

Substitute the values:

\[
z = \frac{12.03 - 12.00}{\frac{0.12}{\sqrt{64}}}
\]
\[
z = \frac{0.03}{\frac{0.12}{8}}
\]
\[
z = \frac{0.03}{0.015} = 2.00
\]

### 3. **Determine the Critical Value and Decision Rule**

For a two-tailed test with a significance level (\(\alpha\)) of 0.05:

- The critical z-values are \(\pm 1.96\).

### 4. **Make a Decision**

- If the computed z-value is greater than 1.96 or less than -1.96, we reject the null hypothesis.
- Here, the computed z-value is 2.00.

Since \(z = 2.00\) is greater than 1.96, we reject the null hypothesis.

### 5. **Conclusion**

There is sufficient evidence at the 0.05 significance level to conclude that the mean content of the cans is different from 12.00 oz.

---

Would you like further details on any part of this process? Here are some related questions you might find interesting:

1. What is the interpretation of the p-value in this context?
2. How would the test change if the sample size were smaller?
3. What are Type I and Type II errors in hypothesis testing?
4. How would we conduct this test using a t-distribution instead of a z-distribution?
5. How can we calculate the confidence interval for the mean content?

**Tip:** In hypothesis testing, always clearly define your null and alternative hypotheses before starting any calculations. This ensures you know exactly what you are testing.

Learn how to test the hypothesis that the mean content of soda cans differs from 12.00 oz using a Z-test. Detailed step-by-step solution with sample mean, standard deviation, and significance level provided. Suitable for advanced high school or college-level students.

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