A study is being made of the failures of an electronic component. There are four types of failures possibly and two mounting positions for the device. The following data have been taken;
FAILURE
A. B. C. D
M. Position 1 22 46 18 9
M. Position 2. 4 17 6 12
Test the hypothesis that the type if failure is independent of the mounting positions ( m. Position). Use a equals 0.01
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To test the hypothesis that the type of failure is independent of the mounting positions, we can perform a chi-square test of independence. The null hypothesis is that the variables are independent, while the alternative hypothesis is that the variables are dependent.
Step 1: Set up the hypotheses:
- Null hypothesis (H0): The type of failure is independent of the mounting positions.
- Alternative hypothesis (H1): The type of failure is dependent on the mounting positions.
Step 2: Set the significance level:
Given that α (significance level) equals 0.01, we will reject the null hypothesis if the p-value is less than 0.01.
Step 3: Calculate the expected frequencies:
To perform the chi-square test, we need to calculate the expected frequencies. The expected frequency for each cell is calculated using the formula:
Expected Frequency = (row total × column total) / grand total
First, let's calculate the row totals and column totals:
FAILURE
A B C D Row Total
M. Position 1 22 46 18 9 95
M. Position 2 4 17 6 12 39
Column Total 26 63 24 21 134
Now, using the expected frequency formula:
- Expected Frequency for Cell A (M.Position 1 + Failure A) = (95 × 26) / 134 ≈ 18.66
- Expected Frequency for Cell B (M.Position 1 + Failure B) = (95 × 63) / 134 ≈ 44.69
- Expected Frequency for Cell C (M.Position 1 + Failure C) = (95 × 24) / 134 ≈ 17.00
- Expected Frequency for Cell D (M.Position 1 + Failure D) = (95 × 21) / 134 ≈ 14.14
- Expected Frequency for Cell A (M.Position 2 + Failure A) = (39 × 26) / 134 ≈ 7.56
- Expected Frequency for Cell B (M.Position 2 + Failure B) = (39 × 63) / 134 ≈ 18.34
- Expected Frequency for Cell C (M.Position 2 + Failure C) = (39 × 24) / 134 ≈ 6.97
- Expected Frequency for Cell D (M.Position 2 + Failure D) = (39 × 21) / 134 ≈ 5.13
Step 4: Calculate the chi-square test statistic:
The chi-square test statistic can be calculated using the formula:
Χ² = ∑ (observed frequency - expected frequency)² / expected frequency
Using the expected frequencies and observed frequencies from the given data, we can calculate the chi-square value:
Χ² = [(22 - 18.66)² / 18.66] + [(46 - 44.69)² / 44.69] + [(18 - 17)² / 17] + [(9 - 14.14)² / 14.14]
+ [(4 - 7.56)² / 7.56] + [(17 - 18.34)² / 18.34] + [(6 - 6.97)² / 6.97] + [(12 - 5.13)² / 5.13]
Step 5: Determine the degrees of freedom:
The degrees of freedom can be calculated using the formula:
Degrees of Freedom = (number of rows - 1) × (number of columns - 1)
In this case, we have (2 - 1) × (4 - 1) = 1 × 3 = 3 degrees of freedom.
Step 6: Find the critical value:
To determine whether to reject or fail to reject the null hypothesis, we need to compare the chi-square test statistic to the critical value from the chi-square distribution table. Using a significance level α = 0.01 and 3 degrees of freedom, the critical value is approximately 11.345.
Step 7: Compare the chi-square test statistic to the critical value:
If the chi-square test statistic is greater than the critical value, we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.
If the calculated chi-square test statistic is greater than 11.345, we reject the null hypothesis.
Step 8: Calculate the p-value:
To calculate the p-value, we compare the calculated chi-square test statistic to the chi-square distribution with the appropriate degrees of freedom. We find the p-value by looking up the chi-square test statistic in the chi-square distribution table.
Step 9: Make a conclusion:
If the p-value is less than the significance level (α), we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.
That's the step-by-step process to test the hypothesis that the type of failure is independent of the mounting positions.
To test the hypothesis that the type of failure is independent of the mounting position, we can perform a chi-square test for independence. This test helps us determine if there is a significant association between categorical variables.
Here's how to conduct the test:
Step 1: Set up the null and alternative hypotheses.
- Null hypothesis (H0): The type of failure is independent of the mounting position.
- Alternative hypothesis (H1): The type of failure is dependent on the mounting position.
Step 2: Determine the expected frequencies.
- Calculate the row totals, column totals, and the total number of observations.
- Use these values to calculate the expected frequency for each cell under the assumption of independence.
Step 3: Calculate the test statistic.
- Use the formula for chi-square test statistic: χ² = Σ (Observed - Expected)² / Expected.
Step 4: Determine the critical value.
- Look up the critical value for the desired level of significance (α) and the degrees of freedom (df).
- The degrees of freedom for a chi-square test of independence are (r - 1) * (c - 1), where r is the number of rows and c is the number of columns.
Step 5: Compare the test statistic with the critical value.
- If the test statistic is greater than the critical value, reject the null hypothesis.
- If the test statistic is less than or equal to the critical value, fail to reject the null hypothesis.
Let's apply these steps to the given data:
Step 1: The null and alternative hypotheses are:
- H0: The type of failure is independent of the mounting position.
- H1: The type of failure is dependent on the mounting position.
Step 2: Calculate the expected frequencies.
- Calculate row totals: (22 + 46 + 18 + 9) = 95 for position 1, and (4 + 17 + 6 + 12) = 39 for position 2.
- Calculate column totals: (22 + 4) = 26 for failure A, (46 + 17) = 63 for failure B, (18 + 6) = 24 for failure C, and (9 + 12) = 21 for failure D.
- Calculate the total number of observations: 95 + 39 = 134.
- Calculate the expected frequency for each cell: (Row total * Column total) / Total number of observations.
Expected frequencies:
A. B. C. D.
1. 26 35 14 20
2. 9 12 5 7
Step 3: Calculate the test statistic.
- Use the formula: χ² = Σ (Observed - Expected)² / Expected.
- Calculate the values for each cell and sum them up.
Observed frequencies:
A. B. C. D.
1. 22 46 18 9
2. 4 17 6 12
Expected frequencies:
A. B. C. D.
1. 26 35 14 20
2. 9 12 5 7
Calculations:
- For cell 1A: (22 - 26)² / 26 ≈ 0.615
- For cell 1B: (46 - 35)² / 35 ≈ 4.114
- For cell 1C: (18 - 14)² / 14 ≈ 0.571
- For cell 1D: (9 - 20)² / 20 ≈ 5.100
- For cell 2A: (4 - 9)² / 9 ≈ 2.111
- For cell 2B: (17 - 12)² / 12 ≈ 4.250
- For cell 2C: (6 - 5)² / 5 = 0.200
- For cell 2D: (12 - 7)² / 7 ≈ 3.857
Sum of (Observed - Expected)² / Expected: ≈ 20.818
Step 4: Determine the critical value.
- The degrees of freedom are (r - 1) * (c - 1) = (2 - 1) * (4 - 1) = 3.
- At α = 0.01 and df = 3, the critical value from the chi-square distribution table is approximately 11.34.
Step 5: Compare the test statistic with the critical value.
- The test statistic (20.818) is greater than the critical value (11.34).
- Therefore, we reject the null hypothesis.
Conclusion: The test results indicate that there is a significant association between the type of failure and the mounting position.