Answer :
### Solution:
#### Step 1: Define the Hypotheses
We start by defining the null and alternative hypotheses:
- Null Hypothesis [tex]\( H_0 \)[/tex]: [tex]\( p = 0.05 \)[/tex] (The true proportion of dented cans is 5%)
- Alternative Hypothesis [tex]\( H_a \)[/tex]: [tex]\( p > 0.05 \)[/tex] (The true proportion of dented cans is greater than 5%)
#### Step 2: Sample Size and Significance Level
We have a sample size [tex]\( n = 100 \)[/tex] and a significance level of [tex]\( \alpha = 0.05 \)[/tex].
#### Step 3: Sample Proportion
The sample proportion of dented cans is given as 5%, or [tex]\( \hat{p} = 0.05 \)[/tex].
#### Step 4: Population Proportion Under the Null Hypothesis
Under the null hypothesis, the population proportion [tex]\( p_0 = 0.05 \)[/tex].
#### Step 5: Standard Deviation of the Sampling Distribution
The standard deviation for the sampling distribution of the sample proportion is calculated using the formula:
[tex]\[ \sigma_{\hat{p}} = \sqrt{\frac{p_0 (1 - p_0)}{n}} \][/tex]
Given the values:
[tex]\[ \sigma_{\hat{p}} = \sqrt{\frac{0.05 \times 0.95}{100}} \approx 0.0218 \][/tex]
#### Step 6: Calculate the Z-Score
The z-score for the sample proportion is calculated as:
[tex]\[ z = \frac{\hat{p} - p_0}{\sigma_{\hat{p}}} \][/tex]
Substituting the given values:
[tex]\[ z = \frac{0.05 - 0.05}{0.0218} = 0 \][/tex]
#### Step 7: Find the Critical Value
For a significance level of [tex]\( \alpha = 0.05 \)[/tex] and a right-tailed test, we need to find the critical value. The critical value for [tex]\( \alpha = 0.05 \)[/tex] is:
[tex]\[ z_{0.05} \approx 1.645 \][/tex]
#### Step 8: Compare the Z-Score to the Critical Value
We compare the calculated z-score to the critical value:
[tex]\[ z = 0 \quad \text{and} \quad z_{0.05} = 1.645 \][/tex]
Since [tex]\( 0 \leq 1.645 \)[/tex], we fail to reject the null hypothesis.
#### Conclusion
Since we fail to reject the null hypothesis, we conclude that there is not enough evidence to suggest that the proportion of dented cans is greater than 5%. Therefore, the manager should continue to run the machines.
### Implications of Errors
1. Type I Error (Rejecting [tex]\( H_0 \)[/tex] when it is true):
- Consequence: The manager will shut down the machines for repair at great cost, unnecessarily.
2. Type II Error (Failing to reject [tex]\( H_0 \)[/tex] when it is false):
- Consequence: The manager will let the machines continue to run, which will continue to damage more than 5% of the cans. Profit will be lost, and the machines will still need to be repaired.
By carefully setting up and interpreting this hypothesis test, the manager can make a more informed decision regarding whether to shut down the machines for repairs or not.
#### Step 1: Define the Hypotheses
We start by defining the null and alternative hypotheses:
- Null Hypothesis [tex]\( H_0 \)[/tex]: [tex]\( p = 0.05 \)[/tex] (The true proportion of dented cans is 5%)
- Alternative Hypothesis [tex]\( H_a \)[/tex]: [tex]\( p > 0.05 \)[/tex] (The true proportion of dented cans is greater than 5%)
#### Step 2: Sample Size and Significance Level
We have a sample size [tex]\( n = 100 \)[/tex] and a significance level of [tex]\( \alpha = 0.05 \)[/tex].
#### Step 3: Sample Proportion
The sample proportion of dented cans is given as 5%, or [tex]\( \hat{p} = 0.05 \)[/tex].
#### Step 4: Population Proportion Under the Null Hypothesis
Under the null hypothesis, the population proportion [tex]\( p_0 = 0.05 \)[/tex].
#### Step 5: Standard Deviation of the Sampling Distribution
The standard deviation for the sampling distribution of the sample proportion is calculated using the formula:
[tex]\[ \sigma_{\hat{p}} = \sqrt{\frac{p_0 (1 - p_0)}{n}} \][/tex]
Given the values:
[tex]\[ \sigma_{\hat{p}} = \sqrt{\frac{0.05 \times 0.95}{100}} \approx 0.0218 \][/tex]
#### Step 6: Calculate the Z-Score
The z-score for the sample proportion is calculated as:
[tex]\[ z = \frac{\hat{p} - p_0}{\sigma_{\hat{p}}} \][/tex]
Substituting the given values:
[tex]\[ z = \frac{0.05 - 0.05}{0.0218} = 0 \][/tex]
#### Step 7: Find the Critical Value
For a significance level of [tex]\( \alpha = 0.05 \)[/tex] and a right-tailed test, we need to find the critical value. The critical value for [tex]\( \alpha = 0.05 \)[/tex] is:
[tex]\[ z_{0.05} \approx 1.645 \][/tex]
#### Step 8: Compare the Z-Score to the Critical Value
We compare the calculated z-score to the critical value:
[tex]\[ z = 0 \quad \text{and} \quad z_{0.05} = 1.645 \][/tex]
Since [tex]\( 0 \leq 1.645 \)[/tex], we fail to reject the null hypothesis.
#### Conclusion
Since we fail to reject the null hypothesis, we conclude that there is not enough evidence to suggest that the proportion of dented cans is greater than 5%. Therefore, the manager should continue to run the machines.
### Implications of Errors
1. Type I Error (Rejecting [tex]\( H_0 \)[/tex] when it is true):
- Consequence: The manager will shut down the machines for repair at great cost, unnecessarily.
2. Type II Error (Failing to reject [tex]\( H_0 \)[/tex] when it is false):
- Consequence: The manager will let the machines continue to run, which will continue to damage more than 5% of the cans. Profit will be lost, and the machines will still need to be repaired.
By carefully setting up and interpreting this hypothesis test, the manager can make a more informed decision regarding whether to shut down the machines for repairs or not.
