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 ? = 0.74 gram. Suppose it is known that for the population of all Anna's hummingbirds, the mean weight is ? = 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 ? = 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?

*a. H*_{0}: *?* = 4.5 g;
*H*_{1}: *?* ? 4.5 g; two-tailed

*b. H*_{0}: *?* = 4.5 g;
*H*_{1}: *?* > 4.5 g;
right-tailed

*c. H*_{0}: *?* = 4.5 g;
*H*_{1}: *?* < 4.5 g; left-tailed

*d. H*_{0}: *?* < 4.5 g;
*H*_{1}: *?* = 4.5 g; left-tailed

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

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

b. The standard normal, since we assume that *x* has a
normal distribution with unknown
*?*.

c. The Student's *t*, since we assume that *x* has
a normal distribution with known *?*.

d. The Student's *t*, since *n* is large with
unknown *?*.

What is the 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 *?*?

a. At the *?* = 0.10 level, we reject the null hypothesis
and conclude the data are statistically significant.

b. At the *?* = 0.10 level, we reject the null hypothesis
and conclude the data are not statistically
significant.

c. At the *?* = 0.10 level, we fail to reject the null
hypothesis and conclude the data are statistically significant.

d. At the *?* = 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.

a. There is sufficient evidence at the 0.10 level to conclude that humming birds in the Grand Canyon weigh less than 4.50 grams.

b. There is insufficient evidence at the 0.10 level to conclude that humming birds in the Grand Canyon weigh less than 4.50 grams.

Answer #1

(a)

c. H0: ? = 4.5 g; H1: ? < 4.5 g; left-tailed

(b)

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

(c)

The test statistics is

The p-value is :

p-value = P(z < -2.48) = 0.0066

(d)

Since p-value is less than 0.10 so we reject the null hypothesis.

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

(e)

a. There is sufficient evidence at the 0.10 level to conclude that humming birds in the Grand Canyon weigh less than 4.50 grams.

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...

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...

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...

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