Question

Allen's hummingbird (Selasphorus sasin) has been studied by zoologist Bill Alther.† Suppose a small group of 13 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 = 0.36 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 largeσ is unknown

-uniform distribution of weights

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

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

-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.

-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.

(d) Find the sample size necessary for an 80% confidence level with a maximal margin of error E = 0.12 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 = 0.36
sample size, n = 13

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 * 0.36/sqrt(13)
ME = 0.13

CI = (xbar - Zc * s/sqrt(n) , xbar + Zc * s/sqrt(n))
CI = (3.15 - 1.28 * 0.36/sqrt(13) , 3.15 + 1.28 * 0.36/sqrt(13))
CI = (3.02 , 3.28)

lower limit: 3.02 ,

upper limit: 3.28

margin of error: 0.13

b)
-σ is known

-normal distribution of weights

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 = 0.12, σ = 0.36

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 * 0.36/0.12)^2
n = 14.75

Therefore, the sample size needed to satisfy the condition n >= 14.75 and it must be an integer number, we conclude that the minimum required sample size is n = 15
Ans : Sample size, n = 15