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 If 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

Question

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 If 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

Answer 1

To determine whether the reported average of 299 miles is statistically different from the observed sample mean, we can perform a hypothesis test using the given information.

Here are the steps to conduct the hypothesis test:

Step 1: State the null and alternative hypotheses:
The null hypothesis (H0) states that there is no significant difference between the reported average (μ) and the observed sample mean (ō̄). H0: μ = 299
The alternative hypothesis (H1) states that there is a significant difference between the reported average and the observed sample mean. H1: μ ≠ 299

Step 2: Select the level of significance:
In this case, the level of significance (α) is given as 0.05, corresponding to a 95% confidence level.

Step 3: Calculate the test statistic:
The test statistic for comparing means in this case is the t-statistic. Since the population standard deviation is unknown, we use the sample standard deviation (s) to estimate it.

t = (ō̄ - μ) / (s / √n)

Here, ō̄ is the observed sample mean (413/6 miles), μ is the reported average (299 miles), s is the sample standard deviation (42.8 miles), and n is the sample size (30 flights).

Substituting the values into the formula, we get:

t = ((413/6) - 299) / (42.8 / √30)
t ≈ 13.03

Step 4: Determine the critical values:
Since this is a two-tail test, we need to find the critical values for a significance level of 0.025 (0.05 divided by 2). Using a t-table or software, the critical t-values for a sample size of 30 and a 0.025 significance level are approximately -2.045 and 2.045.

Step 5: Compare the test statistic with the critical values:
If the absolute value of the calculated t-statistic exceeds the critical values, we reject the null hypothesis. In this case, |t| = 13.03 is much larger than 2.045.

Step 6: Make the decision:
Since the absolute value of the calculated t-statistic is greater than the critical values, we reject the null hypothesis. This means that there is sufficient evidence to support the alternative hypothesis, indicating a significant difference between the reported average of 299 miles and the observed sample mean of 413/6 miles.

Step 7: Interpret the result:
Yes, this tends to cast doubt on the reported average of 299 miles. The observed sample mean of 413/6 miles is significantly different from the reported average, based on the hypothesis test conducted at a 0.05 level of significance.