Test the claim given, identify null hypothesis, alternative hypothesis, test statistic, p-value or critical values, conclusion about the null hypothesis and final conclusion
Environmental concerns often conflict with modern technology, as is the case with birds that pose a hazard to aircraft during takeoff. An environmental group states that incidents of bird strikes are too rare to justify killing the birds. A pilot's group claims that among aborted takeoffs leading to an aircraft going off the end of the runway, 10% are due to bird strikes. A sample data consist of 74 aborted takeoffs in which the aircraft overran the runway. Among those 74 cases, 5 were due to bird strikes. Use a .05 level of significance to test the claim.
Null hypothesis (H0): The proportion of aborted takeoffs due to bird strikes is equal to or less than 10% (p ≤ 0.10).
Alternative hypothesis (H1): The proportion of aborted takeoffs due to bird strikes is greater than 10% (p > 0.10).
Test statistic: Since we are working with proportions, we can use the z-test statistic.
z = (p̂ - p0) / √(p0 * (1 - p0) / n)
where p̂ is the sample proportion, p0 is the hypothesized proportion, and n is the sample size.
Given that among the 74 aborted takeoffs, 5 were due to bird strikes, the sample proportion is:
p̂ = 5 / 74 ≈ 0.0676
Plugging in the values, we get:
z = (0.0676 - 0.10) / √(0.10 * (1 - 0.10) / 74)
P-value: To find the p-value associated with the test statistic, we need to look up the corresponding z-value in the standard normal distribution table. Using a one-sided test, the critical value at a significance level of 0.05 is approximately 1.645.
If the test statistic z is greater than the critical value (1.645), then we will reject the null hypothesis.
Conclusion about the null hypothesis:
If the calculated test statistic z is greater than the critical value (z > 1.645), we will reject the null hypothesis.
Final conclusion:
If the null hypothesis is rejected, then there is evidence to support the pilot's group claim that more than 10% of aborted takeoffs leading to an aircraft going off the end of the runway are due to bird strikes.
To test the claim made by the pilot's group, we need to perform a hypothesis test.
Null Hypothesis (H0): The proportion of aborted takeoffs due to bird strikes is equal to 10% (p = 0.10).
Alternative Hypothesis (H1): The proportion of aborted takeoffs due to bird strikes is NOT equal to 10% (p ≠ 0.10).
Test Statistic:
To test this claim, we will use the proportion test statistic:
Z = (p̂ - p0) / √(p0(1-p0) / n)
where p̂ is the sample proportion, p0 is the hypothesized proportion, and n is the sample size.
In this case, the sample size (n) is 74, and the number of aborted takeoffs due to bird strikes (x) is 5. So, the sample proportion (p̂) is x / n = 5 / 74 ≈ 0.0676.
Now, we can calculate the test statistic:
Z = (0.0676 - 0.10) / √(0.10(1-0.10) / 74)
Calculating this gives us the value of the test statistic Z.
P-value or Critical Values:
Using a 0.05 level of significance, we will compare the test statistic to the critical values from the standard normal distribution or calculate the p-value.
Conclusion about the Null Hypothesis:
If the test statistic falls within the critical region (based on critical values) or the p-value is less than 0.05, we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.
Final Conclusion:
Based on the calculated test statistic and the p-value, we compare them to the critical values. If the test statistic falls within the critical region or if the p-value is less than 0.05, we reject the null hypothesis that the proportion of aborted takeoffs due to bird strikes is 10%. Otherwise, we fail to reject the null hypothesis.
Please note that the actual calculation of the test statistic, critical values, and p-value is missing from the question, so the final conclusion cannot be determined without these values.