Final answer:
To support the airline's claim of a no-show rate less than 6%, a hypothesis test is performed. The test statistic is calculated from the observed sample proportion, and the p-value is obtained from a standard normal distribution to compare against the significance level. The final answer is the p-value rounded to four decimal places.
Step-by-step explanation:
Finding the P-Value
To find the p-value for the no-show rate, we first set up the hypothesis test. The null hypothesis (H0) is that the no-show rate is at least 6% (p >= 0.06), and the alternative hypothesis (H1) is that the no-show rate is less than 6% (p < 0.06). The number of no-shows observed is 18 out of 380 reservations, leading to a sample proportion (p-hat) of 18/380.
We then calculate the test statistic using the formula for a one-proportion z-test:
z = (p-hat - p) / sqrt(p*(1-p)/n)
where p-hat is the sample proportion, p is the hypothesized proportion (0.06), and n is the sample size (380).
Our test statistic comes to z = (18/380 - 0.06) / sqrt(0.06*(1-0.06)/380). After calculations, we obtain a z-value. We use this z-value to find the p-value from a standard normal distribution table or software that gives the probability of observing a z-value as extreme as or more extreme than the calculated one.
Since this is a left-tailed test, we find the cumulative probability up to our z-value, giving us the p-value. This p-value will denote the probability of observing a proportion as extreme as or more extreme than our sample proportion, given that the null hypothesis is true.