The average length of a flight by regional airlines in the United States has been reported as 299 miles. If a simple random sample of 30 flights by regional airlines were to have ȭ = 413/6 miles and s= 42.8 miles, would this tend to cast doubt on the reported average of 299 miles? Use a two-tail test and the 0.05 level of significance in arriving at your answer.
where answer
Null is rejected
To determine whether the reported average of 299 miles can be doubted based on the given sample data, we need to perform a hypothesis test.
Hypotheses:
The null hypothesis (H₀): The population average flight distance is equal to 299 miles.
The alternative hypothesis (H₁): The population average flight distance is not equal to 299 miles.
Level of significance (α): 0.05 (5%)
Test Statistic:
We will use the t-test since the population standard deviation (σ) is not known and we have a sample size (n) less than 30.
t = (x̄ - μ) / (s / sqrt(n))
x̄ = Sample mean = 413/6 miles
μ = Population mean = 299 miles
s = Sample standard deviation = 42.8 miles
n = Sample size = 30
Calculating the test statistic:
t = (413/6 - 299) / (42.8 / sqrt(30))
t = (68.83 - 299) / (42.8 / sqrt(30))
t ≈ (-230.17) / (7.8068)
t ≈ -29.51
Degrees of Freedom:
Since this is a two-tail test, we have n - 1 = 30 - 1 = 29 degrees of freedom.
Critical Values:
Using a t-table or a statistical calculator, with α = 0.05 and df = 29, we find the critical t-values to be approximately -2.045 and 2.045.
Decision:
Since the calculated t-value (-29.51) is much larger than the critical values (-2.045 and 2.045), we reject the null hypothesis.
Conclusion:
Based on the given sample data, we have sufficient evidence to cast doubt on the reported average flight distance of 299 miles. The sample average of 413/6 miles indicates that the population average flight distance is significantly different from 299 miles.