In this scenario, the consumer group is interested in testing whether the average weight of potato chips per bag is less than the advertised weight of 25 grams. The provided hypotheses are:
[tex]\[ H_0: \mu = 25 \][/tex]
[tex]\[ H_a: \mu < 25 \][/tex]
Here are the detailed steps to determine which type of significance test should be used:
### 1. Identify the Nature of the Test:
- The null hypothesis \( H_0 \) states that the population mean \(\mu\) is equal to 25 grams.
- The alternative hypothesis \( H_a \) states that the population mean \(\mu\) is less than 25 grams.
- This is a one-tailed test because the alternative hypothesis is checking for a mean less than 25 grams.
### 2. Determine the Test Type:
- Since the consumer group is testing the sample mean against a known population mean, and they are interested in whether the sample mean is less than the population mean, this sets up a one-sample t-test or z-test.
- The choice between a t-test and a z-test typically depends on whether the population standard deviation is known and the sample size:
- Z-test: If the population standard deviation is known and the sample size is large (generally \( n \geq 30 \)).
- T-test: If the population standard deviation is unknown or the sample size is small.
### 3. Given Information for This Scenario:
- Let's assume the sample size \( n \) and that the population standard deviation (\(\sigma\)) are both known.
- Given a sample size of \( n = 85 \), and a known population standard deviation.
### 4. Standardized Test Statistic:
- Calculate the test statistic using the Z formula:
[tex]\[
z = \frac{\bar{x} - \mu}{\sigma / \sqrt{n}}
\][/tex]
where:
- \(\bar{x}\) is the sample mean.
- \(\mu\) is the population mean.
- \(\sigma\) is the population standard deviation.
- \(n\) is the sample size.
### 5. Significance Level and Critical Value:
- Choose a significance level (\(\alpha\)), commonly \(\alpha = 0.05\).
- The critical value for a one-tailed z-test at \(\alpha = 0.05\) is \(z = -1.645\).
### Conclusion:
Given that the sample size is large and the population standard deviation is known, the appropriate test here would be a one-sample z-test for the mean. This test will help the consumer group determine if there is sufficient evidence to conclude that the mean weight of the potato chips is less than the advertised 25 grams.
