Question

1. The method of tree ring dating gave the following years A.D. for an archaeological excavation site. Assume that the population of x values has an approximately normal distribution.

1264

1257

1243

1306

1268

1316

1275

1317

1275

(a) Use a calculator with mean and standard deviation keys to find the sample mean year x and sample standard deviation s. (Round your answers to the nearest whole number.

x =	A.D.
s =	yr

(b) Find a 90% confidence interval for the mean of all tree ring dates from this archaeological site. (Round your answers to the nearest whole number.)

lower limit	A.D.
upper limit	A.D.

2.

Sherds of clay vessels were put together to reconstruct rim diameters of the original ceramic vessels at the Wind Mountain archaeological site†. A random sample of ceramic vessels gave the following rim diameters (in centimeters).

15.9

13.4

22.1

12.7

13.1

19.6

11.7

13.5

17.7

18.1

(a) Use a calculator with mean and sample standard deviation keys to find the sample mean x and sample standard deviation s. (Round your answers to one decimal place.)

x =	cm
s =	cm

(b) Compute a 98% confidence interval for the population mean μ of rim diameters for such ceramic vessels found at the Wind Mountain archaeological site. (Round your answers to one decimal place.)

lower limit	cm
upper limit	cm

3. 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 = 0.30 gram.

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

n is large

uniform distribution of weights

σ is unknownσ is known

normal distribution of weights

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

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

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

c. 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.

d. 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.

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

(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.)

Answer 1

1) a) = 1280

        s = 27

b) df = 9 - 1 = 8

At 90% confidence level, the critical value is t^* = 1.860

The 90% confidence interval is

Lower limit = 1263

Upper limit = 1297

2)a) = 15.8

       s = 3.5

df = 10 - 1 = 9

At 98% confidence level, the critical value is t^* = 2.821

The 98% confidence interval is

Lower limit = 12.7

Upper limit = 18.9

3) At 80% confidence level, the critical value is z_0.1 = 1.28

The 80% confidence interval is

Lower limit = 3.04

Upper limit = 3.26

Margin of error is

b) is known

Normal distribution of weights.

Option - d) 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) E = 0.16