Statistics Questions

Communter Flight: Normal Distribution

Question 1

With a small plane, the cargo weight (passengers and luggage) is crucial to the efficiency of the flight. Suppose an airline runs a commuter flight that takes 40 passengers. They know that the weights for typical customers on this flight have an approximately normal distribution with a mean of 70 kg and a standard deviation of 10 kg. The weights of typical luggage per person have an approximately normal distribution with a mean of 20 kg and a standard deviation of 3.5 kg. (Ignore the weight of the pilot and the co?pilot).

(i) What type of distribution describes the total weight (W) for a random sample of 40 customers and their luggage? Briefly justify your answer including any assumptions that would be necessary to make.

(ii) Calculate the mean and the standard deviation for (W).

(iii) If the total weight of passengers and their luggage should not exceed 3700 kg, what is the probability that a sold?out flight (40 passengers and their luggage) will exceed the weight limit?

(iv) The pilot and co?pilot both have safety checks to do before the flight. The time it takes them to complete these are independent normally?distributed random variables. For the pilot, the mean time E(P) = 10 minutes with variance Var(P) = 4.5 minutes. For the co?pilot, the mean time E(C) = 8 minutes with variance Var(C) = 1.5 minutes. What is the probability that the copilot finishes his checks before the pilot?

Question 2

(a) The media often report about the dangers of cell phone radiation as a cause of cancer. The Excel file Ass3_q2a contains data from a US environmental working group about radiation emissions (in W/kg) from a sample of cell?phones.

(i) Use the data and the relevant formula to construct a 90% confidence interval estimate of the population mean radiation emission.

(ii) Use Excel’s Descriptive statistics and the method shown in Examples class 8 to find the 90% confidence interval directly, as a check for your answer to part (i). Copy and paste the relevant Excel output into your word document (or print from Excel and attach if you are hand?writing your report).

(iii) Write one or two sentences to explain what the resulting interval suggests about the Federal Communication Commission standard that cell?phone radiation must be 1.6 W/kg or less.

(b) In the Introductory Survey we asked if you had paid employment while studying at university. A summary of the results for this question from the survey is shown in the table below. ‘Working’ denotes students who said that they had some kind of paid employment while studying, ‘Total’ denotes the total number of students who responded to these questions in the surveys.

STAT101 includes a very wide range of students. For this question we will assume that these samples can be regarded as a random sample from the total population of first year UC students in Semester 2 2019.

(i) Use the data to construct a 95% confidence interval for the population proportion of first year UC students who were working while studying at the start of Semester 2 2019.

(ii) Does the confidence interval support the hypothesis that less than half of first year students are also in paid employment? Explain in one or two sentences.