Allen's hummingbird (Selasphorus sasin) has been studied byzoologist Bill Alther.† Suppose a small group of 20 Allen'shummingbirds has been under study in Arizona. The average weightfor these birds is x = 3.15 grams. Based on previous studies, wecan assume that the weights of Allen's hummingbirds have a normaldistribution, with ? = 0.34 gram.

(a) Find an 80% confidence interval for the average weights ofAllen's hummingbirds in the study region. What is the margin oferror? (Round your answers to two decimal places.)

lower limit:

upper limit:

margin of error:

(b) What conditions are necessary for your calculations? (Selectall that apply.)

uniform distribution of weights

? is unknown

n is large

normal distribution of weights

? is known

(c) Interpret your results in the context of this problem.

The probability to the true average weight of Allen'shummingbirds is equal to the sample mean.

The probability that this interval contains the true averageweight of Allen's hummingbirds is 0.80.

The probability that this interval contains the true averageweight of Allen's hummingbirds is 0.20.

There is a 20% chance that the interval is one of the intervalscontaining the true average weight of Allen's hummingbirds in thisregion.

There is an 80% chance that the interval is one of the intervalscontaining the true average weight of Allen's hummingbirds in thisregion.

(d) Find the sample size necessary for an 80% confidence levelwith a maximal margin of error E = 0.09 for the mean weights of thehummingbirds. (Round up to the nearest whole number.)