Allen's hummingbird (Selamphorus sasin) has been studied by zoologist Ather. Sucpose a smal group of 19 Alen's hummingbirds has been under study in Artzona. The average weight for these birds is-3.15 grams. Based on previous studies, we can assume that the weights of Allen's hummingbirds have a normal distribution, wthe0.40 gram. (a) Find an 80 contdence interval for the average weights of Allen's hummingbirds in the study region. What is the margin of emror? (Round your answers to two deomal piaces.) lower imt upper imt margin of erron (b) what concitions are necessary for your calculations? (Select all that apply.) E known Osunknown O nlarge E normal distibution of weighes O unform distribution of weights (c) Irterpret your resuts in the contest of this problem O The probabiity that this interval contains the true average weight of Allen's humminobirds is 080. O The probabilty that this interval contains the true average weight of Allen's hummingbirds s 0.20. O we are 20% confident that the true average weight of Allen's hummingbirds fals within this interval. • ve are s0 confident that the true average weght of Allen's hummingbirds fals within this interval (6) Find the sample size necessary for an o 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
Allen's hummingbird (Selamphorus sasin) has been studied by zoologist Ather. Sucpose a smal group of 19 Alen's hummingbirds has been under study in Artzona. The average weight for these birds is-3.15 grams. Based on previous studies, we can assume that the weights of Allen's hummingbirds have a normal distribution, wthe0.40 gram. (a) Find an 80 contdence interval for the average weights of Allen's hummingbirds in the study region. What is the margin of emror? (Round your answers to two deomal piaces.) lower imt upper imt margin of erron (b) what concitions are necessary for your calculations? (Select all that apply.) E known Osunknown O nlarge E normal distibution of weighes O unform distribution of weights (c) Irterpret your resuts in the contest of this problem O The probabiity that this interval contains the true average weight of Allen's humminobirds is 080. O The probabilty that this interval contains the true average weight of Allen's hummingbirds s 0.20. O we are 20% confident that the true average weight of Allen's hummingbirds fals within this interval. • ve are s0 confident that the true average weght of Allen's hummingbirds fals within this interval (6) Find the sample size necessary for an o 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
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Transcribed Image Text:### Educational Content on Confidence Intervals
#### Study of Allen's Hummingbird (Selasphorus sasin)
A small group of 19 Allen's hummingbirds has been under study in Arizona. The average weight for these birds is \( \bar{x} = 3.15 \) grams. Based on previous studies, the weights of Allen's hummingbirds are assumed to follow a normal distribution with a standard deviation \( \sigma = 0.40 \) gram.
**(a) Confidence Interval Calculation:**
To find an 80% confidence interval for the average weight:
- Lower limit: \( 3.02 \)
- Upper limit: \( 3.27 \)
- Margin of error: \( 0.12 \)
**(b) Conditions for Confidence Interval Calculation:**
The following conditions are necessary:
- \(\sigma\) is known.
- Normal distribution of weights is assumed.
**(c) Interpretation in Context:**
For this problem:
- We are 80% confident that the true average weight of Allen's hummingbirds falls within this interval.
**(d) Sample Size for Margin of Error:**
To find the sample size necessary for an 80% confidence level with a maximal margin of error \( E = 0.06 \) for the mean weight:
- Required sample size: 30 hummingbirds
This study demonstrates the application of confidence intervals in estimating population parameters when certain statistical conditions are met.
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