A gasoline tank for a certain car is designed to hold 15 gallons of gas. Suppose that the actual capacity of a randomly selected tank has a distribution that is approximately Normal with a mean of 15.0 gallons and a standard deviation of 0.15 gallons. If a simple random sample of four tanks is selected, and their capacities can be considered independent, what is the probability that all four will hold between 14.75 and 15.10 gallons of gas?

Answer:Tanks of 15.3081 gallons and larger are considered too large.Step-by-step explanation:Problems of normally distributed samples are solved using the z-score formula.In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the zscore of a measure X is given by:[tex]Z = \frac{X - \mu}{\sigma}[/tex]The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.In this problem, we have that:[tex]\mu = 15, \sigma = 0.15[/tex]How large does a tank have to be to be considered too largelargest 2%, so at least the 98th percentile.The 98th percentile is X when Z = 0.98. So it is X when Z = 2.054.[tex]Z = \frac{X - \mu}{\sigma}[/tex][tex]2.054 = \frac{X - 15}{0.15}[/tex][tex]X - 15 = 2.054*0.15[/tex][tex]X = 15.3081[/tex]Tanks of 15.3081 gallons and larger are considered too large.