Question

Allen's hummingbird (*Selasphorus sasin*) has been
studied by zoologist Bill Alther.† Suppose a small group of 18
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* = 0.22
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.)

uniform distribution of weightsnormal distribution of
weights*σ* is unknown*σ* is known*n* is
large

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

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

(d) Find the sample size necessary for an 80% confidence level with
a maximal margin of error *E* = 0.16 for the mean weights of
the hummingbirds. (Round up to the nearest whole number.)

hummingbirds

(f)Use the Student's *t* distribution to find
*t*_{c} for a 0.95 confidence level when
the sample is 21. (Round your answer to three decimal places.)

Answer #1

Part a)

Confidence Interval :-

X̅ ± Z( α /2) σ / √ ( n )

Z(α/2) = Z (0.2 /2) = 1.282

3.15 ± Z (0.2/2 ) * 0.22/√(18)

Lower Limit = 3.15 - Z(0.2/2) 0.22/√(18)

Lower Limit = 3.08

Upper Limit = 3.15 + Z(0.2/2) 0.22/√(18)

Upper Limit = 3.22

**80% Confidence interval is ( 3.08 , 3.22 )**

Margin of Error = Z (0.2/2 ) * 0.22/√(18) = 0.07

Part b)

normal distribution of weights

*σ* is known

part c)

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.

Part d)

Sample size can be calculated by below formula

n = (( Z(α/2) * σ) / e )2

n = (( Z(0.2/2) * 0.22 ) / 0.16 )2

Critical value Z(α/2) = Z(0.2/2) = 1.2816

n = (( 1.2816 * 0.22 ) / 0.16 )2

**n = 4**

Required sample size at 80% confident is 4.

Part e)

Critical value t(α/2, n-1) = t(0.05 /2, 21-1) = 2.086 ( from t table )

Allen's hummingbird (Selasphorus sasin) has been
studied by zoologist Bill Alther.† Suppose a small group of 18
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 = 0.32
gram.
(a) Find an 80% confidence interval for the average weights of
Allen's hummingbirds in the study region. What is the margin of...

Allen's hummingbird (Selasphorus sasin) has been
studied by zoologist Bill Alther.† Suppose a small group of 12
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 = 0.26
gram.
(a) Find an 80% confidence interval for the average weights of
Allen's hummingbirds in the study region. (Round your answers to
two...

Allen's hummingbird (Selasphorus sasin) has been
studied by zoologist Bill Alther.† Suppose a small group of 10
Allen's hummingbirds has been under study in Arizona. The average
weight for these birds is xbar = 3.15 grams. Based on
previous studies, we can assume that the weights of Allen's
hummingbirds have a normal distribution, with σ = 0.20
gram.
(a) Find an 80% confidence interval for the average weights of
Allen's hummingbirds in the study region. What is the margin of...

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 ? = 0.30
gram.
(a) Find an 80% confidence interval for the average weights of
Allen's hummingbirds in the study region. What is the margin of...

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...

Allen's hummingbird (Selasphorus sasin) has been
studied by zoologist Bill Alther.† Suppose a small group of 14
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...

Allen's hummingbird (Selasphorus sasin) has been studied by
zoologist Bill Alther.† Suppose a small group of 12 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...

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 = 3.15 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...

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 = 0.40 gram. (a) Find an 80% confidence
interval for the average weights of Allen's hummingbirds in the
study region. What is the margin of...

Allen's hummingbird (Selasphorus sasin) has been
studied by zoologist Bill Alther.† Suppose a small group of 14
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 = 0.32
gram.
(a) Find an 80% confidence interval for the average weights of
Allen's hummingbirds in the study region. What is the margin of...

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