The International Air Transport Association surveys business travelers to develop quality ratings for transatlantic gateway airports. The maximum possible rating is 10. Suppose a simple random sample of 50 business travelers is selected and each traveler is asked to provide a rating for the Miami International Airport. The ratings obtained from the sample of 50 business travelers follow. 6 4 6 8 7 7 6 3 3 8 10 4 8 7 8 7 5 9 5 8 4 3 8 5 5 4 4 4 8 4 5 6 2 5 9 9 8 4 8 9 9 5 9 7 8 3 10 8 9 6 Develop a 95% confidence interval estimate of the population mean rating for Miami. (Round your answers to two decimal places.)

Answer:The 95% confidence interval for the population mean rating is (5.73, 6.95). Step-by-step explanation:We start by calculating the mean and standard deviation of the sample:[tex]M=\dfrac{1}{n}\sum_{i=1}^n\,x_i\\\\\\M=\dfrac{1}{50}(6+4+6+. . .+6)\\\\\\M=\dfrac{317}{50}\\\\\\M=6.34\\\\\\s=\sqrt{\dfrac{1}{n-1}\sum_{i=1}^n\,(x_i-M)^2}\\\\\\s=\sqrt{\dfrac{1}{49}((6-6.34)^2+(4-6.34)^2+(6-6.34)^2+. . . +(6-6.34)^2)}\\\\\\s=\sqrt{\dfrac{229.22}{49}}\\\\\\s=\sqrt{4.68}=2.16\\\\\\[/tex]We have to calculate a 95% confidence interval for the mean. The population standard deviation is not known, so we have to estimate it from the sample standard deviation and use a t-students distribution to calculate the critical value. The sample mean is M=6.34. The sample size is N=50. When σ is not known, s divided by the square root of N is used as an estimate of σM: [tex]s_M=\dfrac{s}{\sqrt{N}}=\dfrac{2.16}{\sqrt{50}}=\dfrac{2.16}{7.071}=0.305[/tex] The degrees of freedom for this sample size are: [tex]df=n-1=50-1=49[/tex] The t-value for a 95% confidence interval and 49 degrees of freedom is t=2.01. The margin of error (MOE) can be calculated as: [tex]MOE=t\cdot s_M=2.01 \cdot 0.305=0.61[/tex] Then, the lower and upper bounds of the confidence interval are: [tex]LL=M-t \cdot s_M = 6.34-0.61=5.73\\\\UL=M+t \cdot s_M = 6.34+0.61=6.95[/tex] The 95% confidence interval for the mean is (5.73, 6.95).