Women stereotypically talk more than men do and researchers wondered how much more. Suppose a study attempted to determine the difference in the mean number of words spoken by men or women per day. The results of the study are summarized in the table. Group Population mean Sample size Sample mean Sample standard deviation Standard error estimate women ?w (unknown) ?w= 36 ?⎯⎯⎯w= 14704 ?w=6215 SEw=1036 men ?m (unknown) ?m= 24 ?⎯⎯⎯m= 15022 ?m=7864 SEm=1605   df=41.4224 Assume the conditions are satisfied for a two‑sample ?‑confidence interval. First, determine the positive critical value, ?, for a 95% confidence interval to estimate how many more words women speak each day on average compared to men, ?w−?m. Give your answer precise to at least three decimal places. Determine lower and upper limits of a 95% confidence interval for how many more words speak each day on average compared to men. A negative number would indicate how many fewer words women speak each day on average compared to men. Give your answers precise to the nearest integer number of words.   ?=

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 + ...