We would like to determine if the population mean rating of free apps is higher than the population mean rating of paid apps for all apps available on Google Play. State the alternative hypothesis:     ["mu1 > mu2", "p1 > p2", "mu sub d > 0"]            After carefully exploring the data, we decided to perform a unpooled (Welch’s) t-test.  Consider the following two outputs and answer the question below:  OUTPUT A  ?⎯⎯⎯x¯ s 1. Free 4.78 0.53 2. Paid 4.52 0.13   OUTPUT B  ?⎯⎯⎯x¯  s 1. Free 4.78 0.47 2. Paid 3.49 0.38 Which of the provided R outputs has led to the decision to perform the unpooled (Welch’s) t-test?                       ["output A", "output B", "both"]            After evaluating the p-value, we failed to reject H0, the results are not statistically significant.  What type of error could we have made if we made one? (type 1, type 2, none)

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 + ...