A random samples of 250 bolts manufactured by machine A and 200 bolts manufactured by machine B showed 24 and 10 defective bolts respectively. Test the hypothesis that the machines are showing difference qualities of performance. Use 5 per cent level of significance.
Very good
To test the hypothesis, we will use the two-sample proportion hypothesis test. The null hypothesis (H0) is that the proportion of defective bolts in machines A and B is the same. The alternative hypothesis (Ha) is that the proportion of defective bolts in machines A and B is different.
Let's denote:
pA = proportion of defective bolts in machine A
pB = proportion of defective bolts in machine B
The test statistic for comparing two proportions is the Z-statistic, given by:
Z = (pA - pB) / sqrt((p̂(1-p̂) / nA) + (p̂(1-p̂) / nB))
where:
p̂ = (x1 + x2) / (n1 + n2)
x1 = number of defective bolts in sample 1
x2 = number of defective bolts in sample 2
n1 = size of sample 1
n2 = size of sample 2
Given the information, we have:
x1 = 24
x2 = 10
n1 = 250
n2 = 200
Calculating p̂:
p̂ = (24 + 10) / (250 + 200)
p̂ = 34 / 450
p̂ ≈ 0.0756
Now, let's calculate the Z-statistic:
Z = (pA - pB) / sqrt((p̂(1-p̂) / nA) + (p̂(1-p̂) / nB))
Z = (0.0756 - pB) / sqrt((0.0756(1-0.0756) / 250) + (0.0756(1-0.0756) / 200))
To calculate the critical value, we need to determine the cutoff value for a 5% significance level. Since it is a two-tailed test, we need to divide alpha (0.05) by 2 (0.025) and look for the corresponding Z-value in the Z-table.
The critical Z-value for a 5% significance level (two-tailed test) is approximately ±1.96.
If the calculated Z-value is greater than the critical Z-value (in absolute value), we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.
After comparing the calculated Z-value with the critical Z-value, we can make a conclusion about the hypothesis.
To test the hypothesis that the machines are showing different qualities of performance, we can use a hypothesis test for comparing two proportions. Here's how you can do it step by step:
Step 1: Define the null and alternative hypotheses.
- Null Hypothesis (H0): The machines A and B have the same quality of performance.
- Alternative Hypothesis (H1): The machines A and B have different qualities of performance.
Step 2: Set the significance level (α) to 0.05 (5%).
Step 3: Calculate the sample proportions.
- For machine A: p̂1 = number of defective bolts from machine A / total number of bolts from machine A = 24/250 = 0.096.
- For machine B: p̂2 = number of defective bolts from machine B / total number of bolts from machine B = 10/200 = 0.05.
Step 4: Calculate the test statistic.
- The test statistic for comparing two proportions is the z-test statistic.
- To compute the z-test statistic, use the formula:
z = (p̂1 - p̂2) / √(p̂(1-p̂) / n1 + p̂(1-p̂) / n2)
where p̂ = (x1 + x2) / (n1 + n2), x1 and x2 are the number of defective bolts, and n1 and n2 are the total number of bolts, respectively.
In our case, p̂ = (24 + 10) / (250 + 200) = 0.080.
Plugging in the values:
z = (0.096 - 0.05) / √(0.08(1-0.08) / 250 + 0.08(1-0.08) / 200)
Step 5: Determine the critical value.
- Since the significance level (α) is 0.05, we need to find the critical value for a two-tailed test.
- The critical value can be found using a standard normal distribution table or a z-table. At a significance level of 0.05, the critical value (z_critical) is approximately ±1.96.
Step 6: Make a decision.
- If the calculated test statistic (z) is greater than the critical value (z_critical) or less than negative z_critical, we reject the null hypothesis (H0).
- Otherwise, if the calculated test statistic (z) is within the range of -z_critical to z_critical, we fail to reject the null hypothesis (H0).
In this case, compare the calculated test statistic (z) to the critical value (z_critical) and make a decision based on the outcomes.
Note: The z-test assumes that the sample satisfies the conditions for using a normal distribution approximation.