To investigate the fluid mechanics of swimming, twenty swimmers each swam a specified distance in a water-filled pool and in a pool where the water was thickened with food grade guar gum to create a syrup-like consistency. Velocity, in meters per second, was recorded and the results are given in the table below. Swimmer Velocity (m/s) Water Guar Syrup 1 0.90 0.94 2 0.92 0.98 3 1.00 0.95 4 1.10 1.15 5 1.20 1.24 6 1.25 1.24 7 1.25 1.28 8 1.30 1.30 9 1.35 1.31 10 1.40 1.42 11 1.40 1.41 12 1.50 1.54 13 1.65 1.57 14 1.70 1.70 15 1.75 1.80 16 1.80 1.78 17 1.80 1.81 18 1.85 1.86 19 1.90 1.87 20 1.95 1.95 The researchers concluded that swimming in guar syrup does not change mean swimming speed. Are the given data consistent with this conclusion? Carry out a hypothesis test using a 0.01 significance level. (Use  ?d = ?water − ?guar syrup.) State the appropriate null and alternative hypotheses. H0: ?d < 0 Ha: ?d = 0 H0: ?d = 0 Ha: ?d < 0      H0: ?d = 0 Ha: ?d ≠ 0 H0: ?d = 0 Ha: ?d > 0 H0: ?d ≠ 0 Ha: ?d = 0 Find the test statistic. (Round your answer to two decimal places.) t =  Use technology to find the P-value. (Round your answer to four decimal places.) P-value =  State the conclusion in the problem context. We fail to reject H0. The data do not provide convincing evidence that swimming in guar syrup changes mean swimming speed.We reject H0. The data do not provide convincing evidence that swimming in guar syrup changes mean swimming speed.    We fail to reject H0. The data provide convincing evidence that swimming in guar syrup changes mean swimming speed.We reject H0. The data provide convincing evidence that swimming in guar syrup changes mean swimming speed.

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 + ...