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 = 0.24 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 knownσ is unknownuniform distribution of weightsnormal distribution of weightsn is large

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

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

(d) Find the sample size necessary for an 80% confidence level with a maximal margin of error E = 0.10 for the mean weights of the hummingbirds. (Round up to the nearest whole number.)
hummingbirds

Answer 1

a)
sample mean, xbar = 3.15
sample standard deviation, σ = 0.24
sample size, n = 17

Given CI level is 80%, hence α = 1 - 0.8 = 0.2
α/2 = 0.2/2 = 0.1, Zc = Z(α/2) = 1.28

ME = zc * σ/sqrt(n)
ME = 1.28 * 0.24/sqrt(17)
ME = 0.07

CI = (xbar - Zc * s/sqrt(n) , xbar + Zc * s/sqrt(n))
CI = (3.15 - 1.28 * 0.24/sqrt(17) , 3.15 + 1.28 * 0.24/sqrt(17))
CI = (3.08 , 3.22)

lower limit = 3.08
upper limit = 3.22
margin of error = 0.07

b)
σ is known

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

d)
The following information is provided,
Significance Level, α = 0.2, Margin or Error, E = 0.1, σ = 0.24

The critical value for significance level, α = 0.2 is 1.28.

The following formula is used to compute the minimum sample size required to estimate the population mean μ within the required margin of error:
n >= (zc *σ/E)^2
n = (1.28 * 0.24/0.1)^2
n = 9.44

Therefore, the sample size needed to satisfy the condition n >= 9.44 and it must be an integer number, we conclude that the minimum required sample size is n = 10
Ans : Sample size, n = 10