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 = 0.66 gram.Suppose it is known that for the population of all Anna's hummingbirds, the mean weight is μ = 4.50 grams. Do the data indicate that the mean weight of these birds in this part of the Grand Canyon is less than 4.50 grams? Use α = 0.01.

(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 < 4.5 g; H₁:  μ = 4.5 g; left-tailedH₀: μ = 4.5 g; H₁:  μ > 4.5 g; right-tailed     H₀: μ = 4.5 g; H₁:  μ < 4.5 g; left-tailedH₀: μ = 4.5 g; H₁:  μ ≠ 4.5 g; two-tailed

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

The Student's t, since we assume that x has a normal distribution with known σ.The Student's t, since n is large with unknown σ.     The standard normal, 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.

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.)


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 = 0.01 level, we reject the null hypothesis and conclude the data are statistically significant.At the α = 0.01 level, we reject the null hypothesis and conclude the data are not statistically significant.     At the α = 0.01 level, we fail to reject the null hypothesis and conclude the data are statistically significant.At the α = 0.01 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 0.01 level to conclude that humming birds in the Grand Canyon weigh less than 4.50 grams.There is insufficient evidence at the 0.01 level to conclude that humming birds in the Grand Canyon weigh less than 4.50 grams.    

Answer 1

Solution:

Given:

Sample size = n = 6

Sample mean =

x has a normal distribution and σ = 0.66 gram.

α = 0.01.

We have to test: if the mean weight of these birds in this part of the Grand Canyon is less than 4.50 grams.

Part a) What is the level of significance?

α = 0.01.

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

H₀: μ = 4.5 g; H₁:  μ < 4.5 g; left-tailed

Part 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.

Compute the z value of the sample test statistic.

Part c) Find (or estimate) the P-value.

For left tailed test , P-value is:

P-value = P(Z < z test statistic)

P-value = P(Z < -2.78)

Look in z table for z = -2.7 and 0.08 and find corresponding area.

P( Z< -2.78 ) = 0.0027

thus

P-value = P(Z < -2.78)

P-value =0.0027

Sketch the sampling distribution and show the area corresponding to the P-value.

Part 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?

Decision Rule:
Reject null hypothesis H0, if P-value < 0.01 level of significance, otherwise we fail to reject H0

Since P-value =0.0027 < 0.01 level of significance, we reject null hypothesis H0.

At the α = 0.01 level, we reject the null hypothesis and conclude the data are statistically significant.

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

There is sufficient evidence at the 0.01 level to conclude that humming birds in the Grand Canyon weigh less than 4.50 grams.