The ministry of transport and communication is responsible for the enforcement of airline safety regulations and has become concerned about an apparent increase in accidents and near mishaps involving commercial aircraft. One area of concern is overloading due to passenger luggage exceeding regulated limits. Each passenger on overseas flight is allowed to carry 30 kg of luggage on board. To check adherences to this regulation, ministry officials conducted a spot check on passengers boarding both South African airways and British airways flights from Kenneth Kaunda international airport. The weight of luggage for each was recorded with the following results;

SOUTH AFRICAN AIRWAYS
No of passengers 138
Mean weight in kg 28.48
Standard deviation 9.45
Variance 89.3025

BRITISH AIRWAYS
No of passengers 184
Mean weight in kg 31.65
Standard deviation 7.21
Variance 63.4709

(a) construct 99 percent confidence intervals for the true mean weight of passengers luggage for each airline
( b) is there any reason to believe that either airline is routinely violating luggage regulations?

To construct confidence intervals for the true mean weight of passengers' luggage for each airline, we can use the formula:

Confidence Interval = Mean ± (Critical Value × Standard Deviation / √Sample Size)

(a) For South African Airways:
Sample size (n) = 138
Mean weight (x̄) = 28.48 kg
Standard deviation (s) = 9.45 kg

To construct a 99% confidence interval, we need to find the critical value associated with a 99% confidence level. This critical value can be found using a t-distribution table or a statistical calculator.

Assuming the distribution is approximately normal and the sample size is large enough, we can use the t-distribution. For a 99% confidence level with a sample size of 138, the critical value is approximately 2.626.

Confidence Interval for South African Airways:
Lower Limit = 28.48 - (2.626 × 9.45 / √138)
Upper Limit = 28.48 + (2.626 × 9.45 / √138)

(b) For British Airways:
Sample size (n) = 184
Mean weight (x̄) = 31.65 kg
Standard deviation (s) = 7.21 kg

Similarly, for a 99% confidence level with a sample size of 184, the critical value is approximately 2.624.

Confidence Interval for British Airways:
Lower Limit = 31.65 - (2.624 × 7.21 / √184)
Upper Limit = 31.65 + (2.624 × 7.21 / √184)

Now we can calculate the confidence intervals for both airlines to determine if there is any reason to believe they are violating luggage regulations.

(b) If the confidence intervals for either airline include the regulated limit of 30 kg, there is no strong evidence to suggest that the airline is routinely violating luggage regulations. However, if the confidence intervals do not contain 30 kg, there may be reason to believe that the airline is violating the regulations.

Please note that this analysis assumes the data is representative, collected randomly, and follows a normal distribution. Additionally, we have used the t-distribution for constructing confidence intervals due to the sample size.