**Total Plasma Volume Analysis in Male Firefighters**Total plasma volume is crucial in determining the necessary plasma component for blood replacement therapy in patients undergoing surgery. It is influenced by a person’s overall health and activity level. Consider a sample of 44 male firefighters, with a plasma volume sample mean of \( \bar{x} = 37.5 \) ml/kg. The standard deviation \( \sigma = 7.40 \) ml/kg for blood plasma distribution is considered.### (a) 99% Confidence Interval Calculation**Task:** Find a 99% confidence interval for the population mean blood plasma volume in male firefighters. Calculate the margin of error.- **Lower Limit:**- **Upper Limit:**- **Margin of Error:***Note: Round answers to two decimal places.*### (b) Necessary Conditions for CalculationsSelect all that apply to ensure valid statistical calculations:- [ ] The distribution of volumes is uniform- [ ] The distribution of volumes is normal- [ ] \( \sigma \) is unknown- [ ] \( n \) is large- [ ] \( \sigma \) is known### (c) Results InterpretationChoose the correct interpretation:- [ ] We are 99% confident that the true average blood plasma volume in male firefighters falls within this interval.- [ ] The probability that this interval contains the true average blood plasma volume in male firefighters is 0.99.- [ ] We are 1% confident that the true average blood plasma volume in male firefighters falls within this interval.- [ ] The probability that this interval contains the true average blood plasma volume in male firefighters is 0.01.### (d) Sample Size Determination**Task:** Determine the sample size necessary for a 99% confidence level with a maximal margin of error \( E = 2.80 \) for the mean plasma volume in male firefighters.- **Sample Size:***Note: Round up to the nearest whole number.*

Givensample size(n)=44Mean(x)=37.5standard deviation(σ)=7.40a)Cofidence Interval(c)=99%α=1-0.99=0.01α2=0.005Critical value(zc)=zα2=z0.005=2.5758 (from standard normal table)Margin of error(E)=zc×σnE=2.5758×7.4044E=2.87Confidence Interval(C.I)=x ± EConfidence Interval(C.I)=37.5 ± 2.87Confidence Interval(C.I)=(37.5-2.87 , 37.5+2.87)Confidence Interval =(34.63 , 40.37)Lower limit = 34.63Upper limit = 40.37margin of error(E)=2.87b)conditions are necessary for calculations isn is largeσ is knownc)Interpretation:-we are 99% confident that the true average blood plasma volume in male firefighterfalls within the intervald)Confidence Interval(c)=99%Margin of error(E)=2.80standard deviation(σ)=7.40α=1-0.99=0.01α2=0.005zα2=z0.005=2.5758 (from standard normal table)Sample size(n)=(zα2×σE)2n=(2.5758×7.402.80)2n=46.3n≅46sample size(n)=46