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

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.