Hearing levels or cockpit noise in an aircraft can damage the hearing of pilots who are exposed to this hazard for many hours. Noise level is described as Low=under 88 decibels, Medium=88-91 decibels, High=92^decibels. At a = .05, is the cockpit noise level independent of flight phase? NOISE
Noise Climb Cruise Descent Row Total
Level
Low 6 2 6 14
Medium 18 3 8 29
High 1 3 14 18
Col total 25 8 28 61
To determine if the cockpit noise level is independent of the flight phase, we can perform a chi-squared test of independence.
Here is how you can calculate the chi-squared test statistic and the degrees of freedom:
Step 1: Set the null and alternative hypotheses:
- Null hypothesis (H0): The cockpit noise level is independent of the flight phase.
- Alternative hypothesis (Ha): The cockpit noise level is dependent on the flight phase.
Step 2: Calculate the expected frequencies for each cell assuming independence. To do this, you can use the row and column totals.
Expected frequency = (row total * column total) / grand total
Using the given data, the expected frequencies for each cell are as follows:
Climb Cruise Descent Total
Low 2.87 0.92 10.21 14
Medium 9.69 3.11 16.19 29
High 12.44 3.97 20.59 18
Step 3: Use the chi-squared formula to calculate the test statistic.
chi-squared = Σ [(observed frequency - expected frequency)^2 / expected frequency]
Using the above expected and observed frequencies, calculate the chi-squared value.
Step 4: Determine the degrees of freedom (df) for the test.
df = (number of rows - 1) * (number of columns - 1)
In this case, df = (3 - 1) * (3 - 1) = 4.
Step 5: Look up the critical chi-squared value in the chi-squared table or use a chi-squared calculator.
At a significance level of α = 0.05 and with 4 degrees of freedom, the critical chi-squared value is approximately 9.488.
Step 6: Compare the calculated chi-squared value to the critical chi-squared value.
- If the calculated chi-squared value is greater than the critical chi-squared value, reject the null hypothesis and conclude that the cockpit noise level is dependent on the flight phase.
- If the calculated chi-squared value is less than or equal to the critical chi-squared value, fail to reject the null hypothesis and conclude that there is not enough evidence to suggest a relationship between the cockpit noise level and the flight phase.
Perform the calculations and compare the chi-squared value to the critical chi-squared value to determine if the cockpit noise level is independent of the flight phase.
To determine whether the cockpit noise level is independent of the flight phase, we can perform a chi-squared test for independence. Here are the steps to calculate the chi-squared test statistic:
Step 1: Set up the hypotheses:
- Null hypothesis (H0): Cockpit noise level is independent of flight phase.
- Alternative hypothesis (Ha): Cockpit noise level is not independent of flight phase.
Step 2: Create an observed frequency table:
We already have the observed frequency table:
Climb Cruise Descent Total
Low 6 2 6 14
Medium 18 3 8 29
High 1 3 14 18
Col total 25 8 28 61
Step 3: Calculate the expected frequencies:
To calculate the expected frequencies, we need to assume that the noise level and flight phase are independent. We can calculate the expected frequencies by taking row totals and column totals into account.
Climb Cruise Descent Total
Low (25/61)(14/61) (28/61) 14
Medium (25/61)(8/61) (28/61) 29
High (25/61)(18/61) (28/61) 18
Col total 25 8 28 61
Step 4: Calculate the chi-squared test statistic:
The chi-squared test statistic can be calculated using the formula:
χ^2 = Σ((O - E)^2 / E)
Where:
O = Observed frequency
E = Expected frequency
Let's calculate the chi-squared value:
χ^2 = ((6-14.34)^2 / 14.34) + ((2-4.67)^2 / 4.67) + ((6-16.99)^2 / 16.99) + ((18-9.61)^2 / 9.61) + ((3-5.89)^2 / 5.89) + ((8-15.50)^2 / 15.50) + ((1-7.35)^2 / 7.35) + ((3-5.65)^2 / 5.65) + ((14-20.34)^2 / 20.34) + ((18-7.07)^2 / 7.07) + ((8-13.93)^2 / 13.93) + ((28-38.58)^2 / 38.58)
Step 5: Calculate the degrees of freedom:
The degrees of freedom for a chi-squared test of independence can be calculated using the formula:
df = (number of rows - 1) * (number of columns - 1)
In this case, df = (3 - 1) * (3 - 1) = 4.
Step 6: Determine the critical value:
To determine the critical value for a chi-squared test with a significance level of 0.05 and 4 degrees of freedom, you can refer to a chi-squared distribution table or use a statistical software. The critical value for this test is 9.49.
Step 7: Compare the chi-squared test statistic with the critical value:
If the chi-squared test statistic is greater than the critical value, we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.
In this case, the calculated chi-squared value is greater than the critical value. Therefore, we reject the null hypothesis.
Conclusion: There is sufficient evidence to suggest that cockpit noise level is not independent of flight phase.