Dentists believe that a diet low in sugary foods can reduce the number of cavities in children. Twenty children whose diets are believed to be high in sugar are examined and the mean number of cavities is 3.7 with a standard deviation of 0.4. Ten children whose diets are believed to be low in sugar are examined and the mean number of cavities is 3.6 with a standard deviation of 1.6. Construct a 99% confidence interval for the true difference between the mean numbers of cavities for children whose diets are high in sugar and those whose diets are low in sugar. Assume that the variances of the two populations are the same. Let Population 1 be children whose diets are believed to be high in sugar and Population 2 be children whose diets are believed to be low in sugar. Round the endpoints of the interval to one decimal place, if necessary.

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 + ...