Allen's hummingbird has been studied by zoologist bill Alther. Suppose a small group of 15 Allen's hummingbirds has been under study in Arizona. The average weight for these birds is x=3.15 grams. Based on previous studies, we can assume that the weights of Allen's hummingbirds have a normal distribution, with o= 0.26 gram.  a. Find an 80% confidence interval for the average weights of Allen's hummingbirds in the study region. What is the margin of error? (Round your answer to two decimal places) lower limit? Upper limit? Margin of error?  b. What conditions are necessary for your calculations? c. Find the sample size necessary for an 80% confidence level with a maximum margin of error E=0.09 for the mean weights of the hummingbirds. (Round your answer to the nearest whole number)

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 + ...