Allen's hummingbird (Selasphorus sasin) has been studied by zoologist Bill Alther.† Suppose a small group of 16 Allen's hummingbirds has been under study in Arizona. The average weight for these birds is x = 3.10 grams. Based on previous studies, we can assume that the weights of Allen's hummingbirds have a normal distribution, with σ = 0.30 gram. When finding an 82% confidence interval, what is the critical value for confidence level? (Give your answer to two decimal places.) zc = (a) Find an 82% 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 weights σ is unknown σ is known uniform distribution of weights n is large (c) Interpret your results in the context of this problem. We are 82% confident that the true average weight of Allen's hummingbirds falls within this interval. The probability that this interval contains the true average weight of Allen's hummingbirds is 0.82. We are 18% confident that the true average weight of Allen's hummingbirds falls within this interval. The probability that this interval contains the true average weight of Allen's hummingbirds is 0.18. (d) Which equation is used to find the sample size n for estimating μ when σ is known? n = zc σ E n = zc E σ n = zc E σ 2 n = zc σ E 2 Find the sample size necessary for an 82% confidence level with a maximal margin of error E = 0.14 for the mean weights of the hummingbirds.

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Description: Allen's hummingbird (Selasphorus sasin) has been studied by zoologist Bill Alther.† Suppose a small group of 16 Allen's hummingbirds has been under study in Arizona. The average weight for these birds is x = 3.10 grams. Based on previous studies, we

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Allen's hummingbird (Selasphorus sasin) has been studied by zoologist Bill Alther.† Suppose a small group of 16 Allen's hummingbirds has been under study in Arizona. The average weight for these birds is x = 3.10 grams. Based on previous studies, we can assume that the weights of Allen's hummingbirds have a normal distribution, with σ = 0.30 gram.

When finding an 82% confidence interval, what is the critical value for confidence level? (Give your answer to two decimal places.)

z_c =

(a)

Find an 82% 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 weights
σ is unknown
σ is known
uniform distribution of weights
n is large

(c)

Interpret your results in the context of this problem.

We are 82% confident that the true average weight of Allen's hummingbirds falls within this interval.
The probability that this interval contains the true average weight of Allen's hummingbirds is 0.82.
We are 18% confident that the true average weight of Allen's hummingbirds falls within this interval.
The probability that this interval contains the true average weight of Allen's hummingbirds is 0.18.

(d)

Which equation is used to find the sample size n for estimating μ when σ is known?

n =

z_c σ

n =

z_c E

n =

z_c E

	2

n =

z_c σ

	2

Find the sample size necessary for an 82% confidence level with a maximal margin of error E = 0.14 for the mean weights of the hummingbirds.

### Confidence Interval and Sample Size Estimation for Allen's Hummingbirds

#### (b) Necessary Conditions for Calculations:
Select all conditions that are applicable:
- □ Normal distribution of weights
- □ σ is unknown
- □ σ is known
- □ Uniform distribution of weights
- □ n is large

#### (c) Interpretation of Results:
Select the correct interpretation within the context of this problem:
- ○ We are 82% confident that the true average weight of Allen's hummingbirds falls within this interval.
- ○ The probability that this interval contains the true average weight of Allen's hummingbirds is 0.82.
- ○ We are 18% confident that the true average weight of Allen's hummingbirds falls within this interval.
- ○ The probability that this interval contains the true average weight of Allen's hummingbirds is 0.18.

#### (d) Sample Size Determination:
Choose the correct equation used to find the sample size $ n $ for estimating $ \mu $ when $ \sigma $ is known:
- ○ $ n = \sqrt{\frac{Z_C \sigma}{E}} $
- ○ $ n = \sqrt{\frac{Z_C E}{\sigma}} $
- ○ $ n = \left(\frac{Z_C E}{\sigma}\right)^2 $
- ○ $ n = \left(\frac{Z_C \sigma}{E}\right)^2 $

**Task**: Find the sample size necessary for an 82% confidence level with a maximal margin of error $ E = 0.14 $ for the mean weights of the hummingbirds.

\[ \_\_\_\_\_\_ \, \text{hummingbirds} \]

Features Features Normal distribution is characterized by two parameters, mean (µ) and standard deviation (σ). When graphed, the mean represents the center of the bell curve and the graph is perfectly symmetric about the center. The mean, median, and mode are all equal for a normal distribution. The standard deviation measures the data's spread from the center. The higher the standard deviation, the more the data is spread out and the flatter the bell curve looks. Variance is another commonly used measure of the spread of the distribution and is equal to the square of the standard deviation.

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