The accompanying table contains the data for the amounts​ (in oz) in cans of a certain soda. The cans are labeled to indicate that the contents are

1212

oz of soda. Use the sign test and a

0.010.01

significance level to test the claim that cans of this soda are filled so that the median amount is

12 oz12 oz.

If the median is not

1212

​oz, are consumers being​ cheated?

To test the claim that the median amount of soda in the cans is 12 oz using the sign test, we follow these steps:

### Step 1: Define the Hypotheses
- **Null Hypothesis (\(H_0\))**: The median amount of soda in the cans is 12 oz.
- **Alternative Hypothesis (\(H_1\))**: The median amount of soda in the cans is not 12 oz.

### Step 2: Collect the Data and Apply the Sign Test
- For each can in the sample, compare the amount of soda to 12 oz:
  - Assign a "+" sign if the amount is greater than 12 oz.
  - Assign a "-" sign if the amount is less than 12 oz.
  - Ignore cans with exactly 12 oz.

Let's assume the data from the cans is:

| Can # | Amount (oz) |
|-------|-------------|
| 1     | 12.1        |
| 2     | 11.9        |
| 3     | 12.2        |
| 4     | 11.8        |
| 5     | 12.0        |
| 6     | 12.3        |
| 7     | 11.7        |
| 8     | 12.4        |
| 9     | 12.0        |
| 10    | 12.1        |

### Step 3: Count the Signs
- Ignore values that are exactly 12 oz.
- Count the number of positive signs (cans with more than 12 oz) and negative signs (cans with less than 12 oz).

### Step 4: Determine the Test Statistic
The test statistic is the smaller of the two counts (number of positive signs and number of negative signs).

### Step 5: Determine the Critical Value
Using the significance level (\(\alpha = 0.01\)), check the critical value for the sign test in a table of critical values for small samples.

### Step 6: Decision Rule
- If the test statistic is less than or equal to the critical value, reject the null hypothesis.
- If the test statistic is greater than the critical value, do not reject the null hypothesis.

### Step 7: Conclusion
- If you reject \(H_0\), this suggests the median is not 12 oz.
- If you do not reject \(H_0\), there is insufficient evidence to conclude that the median is different from 12 oz.

### Practical Consideration: Are Consumers Being Cheated?
- If the median is significantly less than 12 oz, consumers might be considered cheated.
- If the median is greater or equal to 12 oz, consumers are not being cheated.

Do you want to calculate the test statistic and critical value together with the data provided?

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**Relative Questions:**
1. How is the sign test different from other non-parametric tests?
2. What happens if the data has many values exactly equal to 12 oz?
3. How would you calculate a p-value for this sign test?
4. How does the significance level (\(\alpha\)) affect the decision rule in hypothesis testing?
5. What are the assumptions behind using the sign test?

**Tip:** The sign test is especially useful when you have small samples or non-normally distributed data, making it a versatile tool in non-parametric statistics.

Learn how to use the sign test to determine if soda cans are filled with a median amount of 12 oz. This statistical test involves hypothesis testing at a significance level of 0.01, analyzing data to ensure consumers are not cheated.

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