Question

Bill Alther is a zoologist who studies Anna’s hummingbird (Calypte anna). A random sample of nine of these birds was caught, weighed, and released in a remote part of the Grand Canyon. The weights observed (in grams) were :

3.7​2.9 ​3.8 ​4.2 ​4.8 ​3.1 ​3.5​ 4.3​ 3.0

Let x be a random variable representing weights of Anna’s hummingbirds in this part of the Grand Canyon. Assuming x has a normal distribution with sample standard deviation 0.648. The population mean is 4.55 grams, according to the mentioned reference.

(Reference: Hummingbirds, K. Long, W. Alther.)

(a) Calculate the Sample Mean of the above mentioned data values.

(b) Do the data indicate that the mean weight of these birds in this part of the Grand Canyon is less than 4.55 grams? Establish the Null and Alternate Hypothesis.

(c) Compute the Test statistics of the sample data using Student’s t-distribution table. Take = 0.05

(d) Find the P-value or range of P-values.

(e) Conclude the test and interpret the result.

Answer 1

Question (a)

Given the weights of the 9 birds in the sample are

3.7​, 2.9, 3.8, ​4, 4,2, 4.8, 3.1, 3.5, 4.3, 3.0

Mean = Sum of the weights / Number of weights

= 33.9 / 3

= 3.7

Sample Mean = 3.7

Sample Standard Deviation S = 0.648

Population Mean = 4.55

Sample size n = 9

Question (b)

Bill Alther is testing whether mean weight of these birds in this part of the Grand Canyon is less than 4.55 grams so that would be Alternate Hypothesis

Null Hypothesis H0 : = 4.55

Alternate Hypothesis H1 : < 4.55

Question (c)

t-test statistic = ( - 0) / (S / )

0 is the Null Hypothesised value which is 4.55 here

t-test statistic = (3.7 - 4.55) / (0.648 / )

= -0.85 / (0.648 /3)

= -0.85 / 0.216

= -3.93519

So the test statistic = -3.93519

Question (d)

Degrees of freedom = n -1

= 8

The p-value for a t-test statistic of -3.93519 and 8 degrees of freedom for a two-tailed test = 0.004325

The p-value for a t-test statistic  = 0.004 rounded to 3 decimal places

Question (e)

Given Significance level = 0.05

Since the p-value of 0.004 is the significance level of 0.05, Bill Alther can reject the Null Hypothesis H0 and cconclude that there is sufficient evidence at 0.05 significance level to support the claim that the mean weight of these birds in this part of the Grand Canyon is less than 4.55 grams