Question

Bill Alther is a zoologist who studies Anna's hummingbird (Calypte anna).† Suppose that in a remote part of the Grand Canyon, a random sample of six of these birds was caught, weighed, and released. The weights (in grams) were as follows.

3.7       2.9       3.8       4.2       4.8       3.1

The sample mean is x = 3.75 grams. Let x be a random variable representing weights of hummingbirds in this part of the Grand Canyon. We assume that x has a normal distribution and σ = 0.80 gram. Suppose it is known that for the population of all Anna's hummingbirds, the mean weight is μ = 4.45 grams. Do the data indicate that the mean weight of these birds in this part of the Grand Canyon is less than 4.45 grams? Use α = 0.10.

(a) What is the level of significance?

State the null and alternate hypotheses. Will you use a left-tailed, right-tailed, or two-tailed test?

-H0: μ = 4.45 g; H1: μ > 4.45 g; right-tailed

-H0: μ < 4.45 g; H1: μ = 4.45 g; left-tailed   

-H0: μ = 4.45 g; H1: μ ≠ 4.45 g; two-tailed

-H0: μ = 4.45 g; H1: μ < 4.45 g; left-tailed

(b) What sampling distribution will you use? Explain the rationale for your choice of sampling distribution.

-The standard normal, since we assume that x has a normal distribution with known σ.

-The Student's t, since we assume that x has a normal distribution with known σ.   

-The standard normal, since we assume that x has a normal distribution with unknown σ.

-The Student's t, since n is large with unknown σ.

(2b) Compute the z value of the sample test statistic. (Round your answer to two decimal places.)

(c) Find (or estimate) the P-value. (Round your answer to four decimal places.)

(2c) Sketch the sampling distribution and show the area corresponding to the P-value.

(d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level α?

-At the α = 0.10 level, we reject the null hypothesis and conclude the data are statistically significant.

-At the α = 0.10 level, we reject the null hypothesis and conclude the data are not statistically significant.   

-At the α = 0.10 level, we fail to reject the null hypothesis and conclude the data are statistically significant.

-At the α = 0.10 level, we fail to reject the null hypothesis and conclude the data are not statistically significant.

(e) State your conclusion in the context of the application.

-There is sufficient evidence at the 0.10 level to conclude that humming birds in the Grand Canyon weigh less than 4.45 grams.

-There is insufficient evidence at the 0.10 level to conclude that humming birds in the Grand Canyon weigh less than 4.45 grams.   

Answer 1

a)
.10 is the level of significance

-H0: μ = 4.45 g; H1: μ < 4.45 g; left-tailed

b)

The standard normal, since we assume that x has a normal distribution with known σ.

2b)

z = (xbar - mu)/(sigma/sqrt(n))
z = (3.75 - 4.45)/(0.8/sqrt(6))
z = -2.14

c)

P-value = 0.0162

d)
-At the α = 0.10 level, we reject the null hypothesis and conclude the data are statistically significant.

e)
-There is sufficient evidence at the 0.10 level to conclude that humming birds in the Grand Canyon weigh less than 4.45 grams.