In a manufacturing plant, the old machine produced widgets with a defect rate of 12%. The boss wants to check if the new machine has a different defect rate. So he took a random of 820 widgets, and found 85 of them were defective. Are these results significant? How do you know? State any and all these are correct conclusions: Reject H0, Do not reject H0, accept H0, data do not support Ha, data support Ha

To determine if the results of the sample are significant, we can conduct a hypothesis test.

Here are the steps to conduct the hypothesis test:

Step 1: State the null hypothesis (H0) and the alternative hypothesis (Ha):
H0: The defect rate of the new machine is the same as the defect rate of the old machine (p = 0.12)
Ha: The defect rate of the new machine is different from the defect rate of the old machine (p ≠ 0.12)

Step 2: Set the significance level (α)
The significance level is the maximum probability of rejecting the null hypothesis when it is true. Commonly used values for α are 0.05 or 0.01.

Step 3: Calculate the test statistic and the p-value
In this case, we are conducting a test for proportions, so we can use the Z-test statistic. The formula for the Z-test statistic is:
Z = (p - P) / sqrt(P * (1 - P) / n)
where p is the sample proportion (defective items/sample size), P is the hypothesized proportion, and n is the sample size.

In this case, p = 85/820 ≈ 0.1037, P = 0.12, and n = 820.
Calculating the Z-test statistic:
Z = (0.1037 - 0.12) / sqrt(0.12 * (1 - 0.12) / 820) ≈ -0.628.

The p-value is the probability of observing a test statistic at least as extreme as the one calculated, assuming the null hypothesis is true. We can use a standard normal distribution table or a statistical software to find the p-value associated with the Z-test statistic.

Step 4: Compare the p-value to the significance level
If the p-value is less than the significance level (α), we reject the null hypothesis (H0). If the p-value is greater than the significance level, we fail to reject the null hypothesis.

In this case, let's assume we are using a significance level of α = 0.05. The critical value associated with this significance level is approximately ±1.96. By comparing the test statistic to the critical values, we can determine if the results are statistically significant.

Since we calculated a negative test statistic (-0.628) and we are testing for a two-tailed alternative hypothesis, we compare the absolute value of the test statistic (-0.628) to the critical value (1.96). If the absolute value of the test statistic is greater than the critical value, we reject the null hypothesis.

In this case, the absolute value of the test statistic (0.628) is less than the critical value (1.96), indicating that we fail to reject the null hypothesis.

Therefore, the correct conclusion is: Do not reject H0. The data do not support Ha.

To determine if the results are significant and whether to accept or reject the null hypothesis (H0), we need to perform a hypothesis test using statistical analysis. In this case, we want to compare the defect rate of the old machine (H0) to the defect rate of the new machine (Ha).

Let's break down the steps to perform the hypothesis test:

Step 1: Define the null and alternative hypotheses:
- Null hypothesis (H0): The defect rate of the old machine is the same as the defect rate of the new machine.
- Alternative hypothesis (Ha): The defect rate of the old machine is different from the defect rate of the new machine.

Step 2: Set the significance level (alpha):
- The significance level, denoted by alpha (α), determines the threshold for the level of evidence needed to reject the null hypothesis. Common values for alpha are 0.05 (5%) or 0.01 (1%).

Step 3: Calculate the test statistic and p-value:
- In this case, since we are comparing two proportions (defect rates), we can use a hypothesis test called a two-proportion z-test.
- The test statistic is calculated by comparing the observed difference in defect rates to the expected difference assuming the null hypothesis is true.
- The p-value is the probability of obtaining a test statistic as extreme or more extreme than the one observed, assuming the null hypothesis is true.

Step 4: Make a decision:
- If the p-value is less than the significance level (alpha), we reject the null hypothesis (H0) in favor of the alternative hypothesis (Ha). The results are considered significant.
- If the p-value is greater than or equal to the significance level (alpha), we fail to reject the null hypothesis (H0). The results are not considered significant.

Now, we can apply these steps to the given scenario:

1. Calculate the observed proportion defect rate for the new machine:
- Observed proportion = 85 / 820 = 0.1037

2. Calculate the expected proportion defect rate for the old machine (assuming H0 is true):
- Expected proportion = 0.12

3. Calculate the test statistic and p-value using a two-proportion z-test:
- With the given sample sizes, you can use the standard formula for calculating the test statistic and p-value.

4. Compare the p-value with the significance level (alpha):
- If the p-value is less than the chosen significance level (e.g., 0.05), you can reject the null hypothesis (H0) in favor of the alternative hypothesis (Ha). These results would be considered significant.
- If the p-value is greater than or equal to the significance level, you fail to reject the null hypothesis (H0), and the results would not be considered significant.

Based on the outcome of the hypothesis test, you can draw one of the following conclusions:

- Reject H0: If the p-value is less than alpha, you reject the null hypothesis (H0), indicating that there is a significant difference in defect rates between the old and new machines. This means the new machine has a different defect rate.

- Do not reject H0: If the p-value is greater than or equal to alpha, you fail to reject the null hypothesis (H0), suggesting that there is not enough evidence to conclude a significant difference in defect rates between the old and new machines. This means the new machine does not have a different defect rate.

- Accept H0: This conclusion is not accurate in hypothesis testing. We either reject or fail to reject the null hypothesis; we do not accept it as true.

- Data do not support Ha: If the p-value is high, we do not have sufficient evidence to support the alternative hypothesis (Ha). This means the data do not support the claim that there is a difference in defect rates between the old and new machines.

- Data support Ha: If the p-value is low, we would have sufficient evidence to support the alternative hypothesis (Ha). However, this would suggest a significant difference in defect rates between the old and new machines.

Note: The specific conclusion depends on the calculations and the chosen significance level.