The engineers at Ryobi have developed a new and faster charger for their 40-volt batteries used for their cordless lawn mowers. The engineers claim the new charger will charge a discharged battery in less than an average time of 68 minutes. Vince Lawn decides to test this claim and purchases the charger and a new 40-volt battery for his cordless lawn mower. He uses his cordless lawn mower 38 times throughout the year to cut his lawn and each time he makes sure the battery is completely discharged before he puts it on the charger. He finds that the mean time it takes the battery to fully recharge is 64 minutes with a standard deviation of 10.9 minutes. Using Vince Lawn's results, test the engineers claim at the 0.01 level of significance. (Show all steps.)

Definition Definition Measure of central tendency that is the average of a given data set. The mean value is evaluated as the quotient of the sum of all observations by the sample size. The mean, in contrast to a median, is affected by extreme values. Very large or very small values can distract the mean from the center of the data. Arithmetic mean: The most common type of mean is the arithmetic mean. It is evaluated using the formula: μ = 1 N ∑ i = 1 N x i Other types of means are the geometric mean, logarithmic mean, and harmonic mean. Geometric mean: The nth root of the product of n observations from a data set is defined as the geometric mean of the set: G = x 1 x 2 ... x n n Logarithmic mean: The difference of the natural logarithms of the two numbers, divided by the difference between the numbers is the logarithmic mean of the two numbers. The logarithmic mean is used particularly in heat transfer and mass transfer. ln x 2 − ln x 1 x 2 − x 1 Harmonic mean: The inverse of the arithmetic mean of the inverses of all the numbers in a data set is the harmonic mean of the data. 1 1 x 1 + 1 x 2 + ...