Welcome to Westonci.ca, the place where your questions are answered by a community of knowledgeable contributors. Discover detailed answers to your questions from a wide network of experts on our comprehensive Q&A platform. Experience the convenience of finding accurate answers to your questions from knowledgeable experts on our platform.
Sagot :
To answer this question, we need to conduct a hypothesis test comparing the proportions of defective chips from two different plants.
Step 1: State the Hypotheses
We are given:
- [tex]\( p_A \)[/tex]: the true proportion of defective chips from plant [tex]\( A \)[/tex]
- [tex]\( p_B \)[/tex]: the true proportion of defective chips from plant [tex]\( B \)[/tex]
The hypotheses can be stated as:
- [tex]\( H_0: p_A = p_B \)[/tex]
- [tex]\( H_a: p_A > p_B \)[/tex]
Step 2: Collect Sample Data
From the problem:
- Sample size from plant [tex]\( A \)[/tex], [tex]\( n_A = 80 \)[/tex]
- Defective chips from plant [tex]\( A \)[/tex], [tex]\( x_A = 12 \)[/tex]
- Sample size from plant [tex]\( B \)[/tex], [tex]\( n_B = 90 \)[/tex]
- Defective chips from plant [tex]\( B \)[/tex], [tex]\( x_B = 10 \)[/tex]
Step 3: Calculate Sample Proportions
Calculate the sample proportions of defective chips for each plant:
[tex]\[ \hat{p}_A = \frac{x_A}{n_A} = \frac{12}{80} = 0.15 \][/tex]
[tex]\[ \hat{p}_B = \frac{x_B}{n_B} = \frac{10}{90} = 0.1111111111111111 \][/tex]
Step 4: Calculate the Pooled Proportion
The pooled proportion [tex]\(\hat{p}_{\text{pooled}}\)[/tex] is calculated as:
[tex]\[ \hat{p}_{\text{pooled}} = \frac{x_A + x_B}{n_A + n_B} = \frac{12 + 10}{80 + 90} = \frac{22}{170} = 0.12941176470588237 \][/tex]
Step 5: Calculate the Standard Error
The standard error (SE) for the difference in proportions is given by:
[tex]\[ SE = \sqrt{\hat{p}_{\text{pooled}} (1 - \hat{p}_{\text{pooled}}) \left(\frac{1}{n_A} + \frac{1}{n_B}\right)} = \sqrt{0.12941176470588237 \cdot 0.8705882352941176 \left(\frac{1}{80} + \frac{1}{90}\right)} = 0.05157645508324752 \][/tex]
Step 6: Calculate the Z-Score
We calculate the z-score using the sample proportions and the standard error:
[tex]\[ z = \frac{\hat{p}_A - \hat{p}_B}{SE} = \frac{0.15 - 0.1111111111111111}{0.05157645508324752} = 0.7540046873349452 \][/tex]
Step 7: Determine the P-Value
Since we are conducting a one-tailed test (to see if [tex]\( p_A \)[/tex] is greater than [tex]\( p_B \)[/tex]), we need the area to the right of the z-score on the normal distribution:
[tex]\[ p\text{-value} = 1 - \Phi(z) \][/tex]
Using the z-score of 0.7540046873349452:
The corresponding p-value is 0.22542320358226453.
Step 8: Make a Decision
Given that our p-value (0.22542320358226453) is greater than any common significance level (like 0.05 or 0.01), we do not have sufficient evidence to reject the null hypothesis.
Thus, the correct p-value for the given hypotheses is approximately 0.23.
Step 1: State the Hypotheses
We are given:
- [tex]\( p_A \)[/tex]: the true proportion of defective chips from plant [tex]\( A \)[/tex]
- [tex]\( p_B \)[/tex]: the true proportion of defective chips from plant [tex]\( B \)[/tex]
The hypotheses can be stated as:
- [tex]\( H_0: p_A = p_B \)[/tex]
- [tex]\( H_a: p_A > p_B \)[/tex]
Step 2: Collect Sample Data
From the problem:
- Sample size from plant [tex]\( A \)[/tex], [tex]\( n_A = 80 \)[/tex]
- Defective chips from plant [tex]\( A \)[/tex], [tex]\( x_A = 12 \)[/tex]
- Sample size from plant [tex]\( B \)[/tex], [tex]\( n_B = 90 \)[/tex]
- Defective chips from plant [tex]\( B \)[/tex], [tex]\( x_B = 10 \)[/tex]
Step 3: Calculate Sample Proportions
Calculate the sample proportions of defective chips for each plant:
[tex]\[ \hat{p}_A = \frac{x_A}{n_A} = \frac{12}{80} = 0.15 \][/tex]
[tex]\[ \hat{p}_B = \frac{x_B}{n_B} = \frac{10}{90} = 0.1111111111111111 \][/tex]
Step 4: Calculate the Pooled Proportion
The pooled proportion [tex]\(\hat{p}_{\text{pooled}}\)[/tex] is calculated as:
[tex]\[ \hat{p}_{\text{pooled}} = \frac{x_A + x_B}{n_A + n_B} = \frac{12 + 10}{80 + 90} = \frac{22}{170} = 0.12941176470588237 \][/tex]
Step 5: Calculate the Standard Error
The standard error (SE) for the difference in proportions is given by:
[tex]\[ SE = \sqrt{\hat{p}_{\text{pooled}} (1 - \hat{p}_{\text{pooled}}) \left(\frac{1}{n_A} + \frac{1}{n_B}\right)} = \sqrt{0.12941176470588237 \cdot 0.8705882352941176 \left(\frac{1}{80} + \frac{1}{90}\right)} = 0.05157645508324752 \][/tex]
Step 6: Calculate the Z-Score
We calculate the z-score using the sample proportions and the standard error:
[tex]\[ z = \frac{\hat{p}_A - \hat{p}_B}{SE} = \frac{0.15 - 0.1111111111111111}{0.05157645508324752} = 0.7540046873349452 \][/tex]
Step 7: Determine the P-Value
Since we are conducting a one-tailed test (to see if [tex]\( p_A \)[/tex] is greater than [tex]\( p_B \)[/tex]), we need the area to the right of the z-score on the normal distribution:
[tex]\[ p\text{-value} = 1 - \Phi(z) \][/tex]
Using the z-score of 0.7540046873349452:
The corresponding p-value is 0.22542320358226453.
Step 8: Make a Decision
Given that our p-value (0.22542320358226453) is greater than any common significance level (like 0.05 or 0.01), we do not have sufficient evidence to reject the null hypothesis.
Thus, the correct p-value for the given hypotheses is approximately 0.23.
Thank you for choosing our service. We're dedicated to providing the best answers for all your questions. Visit us again. We hope you found this helpful. Feel free to come back anytime for more accurate answers and updated information. Get the answers you need at Westonci.ca. Stay informed by returning for our latest expert advice.