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 = 0.96 gram. Suppose it is known that for the population of all Anna's hummingbirds, the mean weight is μ = 4.80 grams. Do the data indicate that the mean weight of these birds in this part of the Grand Canyon is less than 4.80 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 = 4.8 g; H1: μ > 4.8 g; right-tailed
-H0: μ = 4.8 g; H1: μ ≠ 4.8 g; two-tailed
-H0: μ = 4.8 g; H1: μ < 4.8 g; left-tailed
-H0: μ < 4.8 g; H1: μ = 4.8 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 n is large with unknown σ.
- The standard normal, since we assume that x has a normal distribution with unknown σ.
-The Student's t, since we assume that x has a normal distribution with known 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 4.80 grams.
-There is insufficient evidence at the 0.01 level to conclude that humming birds in the Grand Canyon weigh less than 4.80 grams.
a)
0.01. is the level of significance
H0: μ = 4.8 g; H1: μ < 4.8 g; left-tailed
b)
- The standard normal, since we assume that x has a normal
distribution with known σ.
Test statistic,
z = (xbar - mu)/(sigma/sqrt(n))
z = (3.75 - 4.8)/(0.96/sqrt(6))
z = -2.68
c)
P-value = 0.0037
d)
At the α = 0.01 level, we reject the null hypothesis and conclude the data are statistically significant
e)
-There is sufficient evidence at the 0.01 level to conclude that humming birds in the Grand Canyon weigh less than 4.80 grams.
Get Answers For Free
Most questions answered within 1 hours.