Question

Asked Jul 15, 2019

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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 = 1.00 gram. Suppose it is known that for the population of all Anna's hummingbirds, the mean weight is μ = 4.75 grams. Do the data indicate that the mean weight of these birds in this part of the Grand Canyon is less than 4.75 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?

*H*_{0}: μ < 4.75 g; *H*_{1}: μ = 4.75 g; left-tailed*H*_{0}: μ = 4.75 g; *H*_{1}: μ < 4.75 g; left-tailed *H*_{0}: μ = 4.75 g; *H*_{1}: μ ≠ 4.75 g; two-tailed*H*_{0}: μ = 4.75 g; *H*_{1}: μ > 4.75 g; right-tailed

(b)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.)

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

(b)Compute the

(c) Find (or estimate) the

Step 1

**(a) Level of significance for the test:**

Level of significance (*α*) defines the probability of rejecting the null hypothesis when it is actually true.

It is given that to test whether the mean weight of the birds in the part of the Grand Canyon is less 4.75 grams, level of significance, *α* = 0.01.

Thus, here the level of significance (*α*) is **0.01.**

**Null and alternative hypotheses:**

*Null hypothesis:*

*H*_{0}: *µ* = 4.75

That is, the mean weight of the birds in the part of the Grand Canyon is equal to 4.75 grams.

*Alternative hypothesis:*

*H*_{1}: *µ* < 4.75

mean weight of the birds in the part of the Grand Canyon is less than 4.75 grams.

Since the alternative hypothesis states *µ* < 4.75, it is a L**eft-tailed test.**

Step 2

**(b) Sampling distribution of the test:**

It is given that *σ* = 1.00.

Since the population standard deviation for the test is known, the appropriate test for this case is **one sample** ** z-test**.

**Test statistic for z-test:**

Here, the sample mean, *x*-bar is 3.75.

Population mean, *µ* is 4.75.

Population standard deviation, *σ* = 1.00.

Sample size, *n* is 6.

The test statistic for *z*-test is calculated as **−2.45 **from the calculations given below.

Step 3

**(c) Computation of P-value:**

The *P*-value for the *z*-test can be obtained using the excel formula, “=1-Z.TEST(A1:A...

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