Question

Allen's hummingbird (Selasphorus sasin) has been studied by zoologist Bill Alther.† Suppose a small group of 17 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 σ = 0.40 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 answers to two decimal places.)

lower limit

upper limit

margin of error

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

σ is known

normal distribution of weights

n is large

uniform distribution of weights

σ is unknown

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

a) There is an 80% chance that the interval is one of the intervals containing the true average weight of Allen's hummingbirds in this region.

b)There is a 20% chance that the interval is one of the intervals containing the true average weight of Allen's hummingbirds in this region.

c)The probability to the true average weight of Allen's hummingbirds is equal to the sample mean.

d)The probability that this interval contains the true average weight of Allen's hummingbirds is 0.20.

e)The probability that this interval contains the true average weight of Allen's hummingbirds is 0.80.

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

Answer 1

a)

80% confidence interval for is

- Z/2 * / sqrt(n) < < + Z/2 * / sqrt(n)

3.15 - 1.2816 * 0.40 / sqrt(17) < < 3.15 + 1.2816 * 0.40 / sqrt(17)

3.03 < < 3.27

80% confidence interval is (3.03 , 3.27)

Lower limit = 3.03

Upper limit = 3.27

Margin of error = Z/2 * / sqrt(n)

= 1.2816 * 0.40 / sqrt(17)

= 0.12

b)

We assume that normal distribution of weights.

c)

There is an 80% chance that the interval is one of the intervals containing the true average weight of

Allen's hummingbird in the region.

d)

Sample size = (Z/2 * / E)²

= (1.2816 * 0.40 / 0.09)²

= 32.44

Sample size = 33 (Rounded up to nearest integer)