Q 8.18. - Elementary Statistics

Question

Cottonmouth Litter Size. In the article "The Eastern Cottonmouth (Agkistrodon piscivorus) at the Northern Edge of Its Range" (Journal of Herpetology, Vol. $29$ No. $3$ pp. $391-398$), C. Blem and L. Blem examined the reproductive characteristics of the eastern cottonmouth, a once widely distributed snake whose numbers have decreased recently due to encroachment by humans. A simple random sample of $44$ female cottonmouths yielded the following data on the number of young per litter. 

a. Use the data to obtain a point estimate for the mean number of young per litter, $\mu$ of all female eastern cottonmouths. (Note: $\left.\Sigma x_i=334.\right)$

b. Is your point estimate in part (a) likely to equal $\mu$ exactly? Explain your answer.

Step-by-Step Solution

Answer

Part (a) Estimated value of mean no. of young per litter for the population is $7$

Part (b) No.

Part (a) Step 1: Given information

5	12	7	7	6	8	12	9	7
4	9	6	12	7	5	6	10	3
10	8	8	12	5	6	10	11	3
8	4	5	7	6	11	7	6	8
8	14	8	7	11	7	5	4

Part (a) Step 2: Concept

The formula used: Sample mean $=\frac{1}{n}\times \sum _{i=1}^{44}{x}_i$

Part (a) Step 3: Calculation

Here the sample size is 44 i.e., $n=44$

Let $x_i$ be the $i-t h$ sample observation on number of young per litter. $i=1,2\dots \dots ,44$

Here $\sum _{i=1}^{44}{x}_i=334$

$\therefore$ Sample mean $=\frac{1}{n}\times \sum _{i=1}^{44}{x}_i$

$$\begin{gathered} =\frac{1}{44}\times 334 \\ =7.59 \end{gathered}$$

Mean no of young per litter is $8$ (approximately)

$\therefore$ the point estimate of the population means $=8$

The estimated value of mean no. of young per litter for the population is $7$

Part (b) Step 1: Explanation

No, it's doubtful that the sample mean $\overline{x}$ will be exactly the same as the population mean $\mu$ We're attempting to guess (estimate) the population mean value solely by drawing a sample, or a subset of the population. As a result, some sampling error is unavoidable.