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 = 4.35 grams. Do the data indicate that the mean weight of these birds in this part of the Grand Canyon is less than 4.35 grams? Use α = 0.05.

(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 = 4.35 g; H₁: μ < 4.35 g; left-tailedH₀: μ = 4.35 g; H₁: μ > 4.35 g; right-tailed    H₀: μ < 4.35 g; H₁: μ = 4.35 g; left-tailedH₀: μ = 4.35 g; H₁: μ ≠ 4.35 g; two-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 unknown σ.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 Student's t, since n is large 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 = 0.05 level, we reject the null hypothesis and conclude the data are statistically significant.At the α = 0.05 level, we reject the null hypothesis and conclude the data are not statistically significant.    At the α = 0.05 level, we fail to reject the null hypothesis and conclude the data are statistically significant.At the α = 0.05 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 0.05 level to conclude that humming birds in the Grand Canyon weigh less than 4.35 grams.There is insufficient evidence at the 0.05 level to conclude that humming birds in the Grand Canyon weigh less than 4.35 grams.

Answer 1

a.

The level of significance is 0.05.

Ho: μ=4.35

Ha: μ < 4.35

We will use a left tailed test.

b.

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

the z value of the sample test statistic is -2.227

c.

The p-value is 0.013.

d.

At the α = 0.05 level, we reject the null hypothesis and conclude the data are statistically significant.

e.

There is sufficient evidence at the 0.05 level to conclude that humming birds in the Grand Canyon weigh less than 4.35 grams

SOLUTION

The sample mean is and the known population standard deviation is σ=0.66, and the sample size is

n = 6.

(1) Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

Ho: μ=4.35

Ha: μ < 4.35

This corresponds to a left-tailed test, for which a z-test for one mean, with known population standard deviation will be used.

(2) Rejection Region

The significance level is α=0.05, and the critical value for a left-tailed test is zc​ = −1.64.

The rejection region for this left-tailed test is R = { z : z < −1.64}

(3) Test Statistics

The z-statistic is computed as follows:

(4) Decision about the null hypothesis

Since it is observed that z = −2.227 < zc ​= −1.64, it is then concluded that the null hypothesis is rejected.

Using the P-value approach: The p-value is p = 0.013, and since p = 0.013 < 0.05, it is concluded that the null hypothesis is rejected.

(5) Conclusion

It is concluded that the null hypothesis Ho is rejected. Therefore, there is enough evidence to claim that the population mean μ is less than 4.35, at the 0.05 significance level.

Therefore, there is enough evidence to claim that the mean weight of these birds in this part of the Grand Canyon is less than 4.35 grams.

Confidence Interval

The 95% confidence interval is 3.222<μ<4.278.