Bill is a zoologist who studies Anna’s hummingbirds. 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:
3.7 2.9 3.8 4.2 4.8 3.1
Let x be a random variable representing weight of Anna’s hummingbirds in this part of Grand Canyon. Assume that it has a normal distribution and 𝜎 = 0.70 grams. It is known that for the population of all Anna’s hummingbirds, the mean is 𝜇 = 4.55. Does this sample data indicate that the mean weight of these birds in this part of Grand Canyon is less than 4.55 grams? Use 𝛼 = 0.01.
The following null and alternative hypotheses need to be tested:
"H_0:\\mu\\geq4.55"
"H_1:\\mu<4.55"
This corresponds to a left-tailed test, for which a z-test for one mean, with known population standard deviation will be used.
Based on the information provided, the significance level is "\\alpha = 0.01," and the critical value for a left-tailed test is "z_c = -2.3263."
The rejection region for this left-tailed test is "R = \\{z: z < -2.3263\\}."
The z-statistic is computed as follows:
Since it is observed that"z = -2.7994 < -2.3263=z_c ," it is then concluded that the null hypothesis is rejected.
Using the P-value approach:
The p-value is "p=P(Z<-2.7994)=0.00256," and since "p=0.00256<0.01=\\alpha," it is concluded that the null hypothesis is rejected.
Therefore, there is enough evidence to claim that the mean weight of these birds in this part of Grand Canyon is less than "4.55," at the "\\alpha = 0.01" significance level.
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