Question

Allen's hummingbird (Selasphorus sasin) has been studied by zoologist Bill Alther.† Suppose a small group of 19 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 = 0.32 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.)

normal distribution of weightsσ is knownuniform distribution of weightsn is largeσ is unknown

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

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.The probability that this interval contains the true average weight of Allen's hummingbirds is 0.80.    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.The probability to the true average weight of Allen's hummingbirds is equal to the sample mean.The probability that this interval contains the true average weight of Allen's hummingbirds is 0.20.

(d) Find the sample size necessary for an 80% confidence level with a maximal margin of error E = 0.06 for the mean weights of the hummingbirds. (Round up to the nearest whole number.)
hummingbirds

Answer 1

Given:

Mean, = 3.15 grams

SD, = 0.32 grams

Sample size, n = 19

(a) Corresponding to 80% confidence interval, the critical z value = 1.282

Margin of error , E = = = 0.094

lower limit = = 3.056 ≈ 3.06 grams

upper limit = = 3.244 ≈ 3.24 grams

Margin of error = 0.09

(b) The conditions necessary for calculation are:

Normal distribution of weights

is known

(c) Interpretation:

The probability that this interval contains the true average weight of

Allen's hummingbirds is 0.80

(d) Let the required sample size be n

We should have E ≤ 0.06

-> = ≤ 0.06

-> n ≥ 46.75

Thus, the sample size of 47 is necessary for the required condition to be met