How to solve this?
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
new machine, defect rate is 85/820 = .1037 or 1037%
draw you conclusion from that
To determine if the results are significant and make conclusions about the hypothesis, we need to conduct a hypothesis test. Here's how:
Step 1: State the null and alternative hypotheses (H0 and Ha):
- Null Hypothesis (H0): The new machine has the same defect rate as the old machine.
- Alternative Hypothesis (Ha): The new machine has a different defect rate than the old machine.
Step 2: Choose a significance level (α), which represents the probability of rejecting H0 when it is true. Commonly used values for α are 0.05 and 0.01.
Step 3: Calculate the test statistic. In this case, since we are comparing two proportions (defect rates), we will use the z-test for proportions.
The formula for the z-test statistic is:
z = (p̂1 - p̂2) / √((p̂(1-p̂))/n1 + (p̂(1-p̂))/n2)
where p̂1 is the defect rate of the old machine, p̂2 is the defect rate of the new machine, n1 is the sample size of the old machine, and n2 is the sample size of the new machine.
We need to calculate p̂1, p̂2, and plug in the values in the formula.
p̂1 = defect rate of the old machine = 0.12
p̂2 = (number of defective widgets in the sample of the new machine) / (sample size of the new machine) = 85/820
n1 = sample size of the old machine = total sample size of the old machine (unknown)
n2 = sample size of the new machine = 820
Step 4: Calculate the critical value or p-value. The critical value is obtained from the standard normal distribution table for the given significance level. If the calculated z-test statistic is greater than the critical value (or if the p-value is less than α), we reject the null hypothesis.
Step 5: Make a conclusion and state the correct conclusion(s) from the options provided.
Following these steps, you can conduct the hypothesis test and determine if the results are significant and what conclusions can be made.