Question 1185509: Allen's hummingbird (Selasphorus sasin) has been studied by zoologist Bill Alther.† Suppose a small group of 11 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.
When finding an 80% confidence interval, what is the critical value for confidence level? (Give your answer to two decimal places.)
zc =
(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
Answer by CPhill(1987)   (Show Source):
You can put this solution on YOUR website! Here's how to calculate the 80% confidence interval and margin of error:
1. **Find the critical value (z_c):**
For an 80% confidence level, the alpha (α) is 1 - 0.80 = 0.20. Since confidence intervals are two-tailed, we divide alpha by 2: 0.20 / 2 = 0.10.
We want to find the z-score that corresponds to an area of 0.10 in *each* tail of the standard normal distribution. This means the area to the *left* of the z-score is 1 - 0.10 = 0.90. Using a standard normal (z) table or calculator, look up the z-score corresponding to a cumulative area of 0.90. You'll find that z_c ≈ 1.28.
So, z_c = 1.28
2. **Calculate the margin of error (E):**
The margin of error is calculated as:
E = z_c * (σ / √n)
Where:
* z_c is the critical value (1.28)
* σ is the population standard deviation (0.36 gram)
* n is the sample size (11 hummingbirds)
E = 1.28 * (0.36 / √11)
E ≈ 1.28 * (0.36 / 3.3166)
E ≈ 1.28 * 0.1085
E ≈ 0.14 gram (rounded to two decimal places)
3. **Calculate the confidence interval:**
The confidence interval is calculated as:
(x̄ - E, x̄ + E)
Where x̄ is the sample mean (3.15 grams).
Lower Limit = x̄ - E = 3.15 - 0.14 = 3.01 grams
Upper Limit = x̄ + E = 3.15 + 0.14 = 3.29 grams
Therefore:
* z_c = 1.28
* Lower limit = 3.01 grams
* Upper limit = 3.29 grams
* Margin of error = 0.14 grams
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