Q 8.20. - Elementary Statistics

Question

Cottonmouth Litter Size. Refer to Exercise 8.18. Assume that $\sigma =2.4$

a. Obtain a $95\%$ confidence interval for the mean number of young per litter of all-female eastern cottonmouths.

b. Interpret your result in part (a).

c. Does the mean number of a young per litter of all-female eastern cottonmouths lie in the confidence interval you obtained in part (a)? Explain your answer.

Step-by-Step Solution

Answer

Part (a) For all-female eastern cottonmouths, the $95$ percent confidence interval for the mean number of young per litter is $(6.9, 8.3)$

Part (b) In all-female eastern cottonmouth litters, the average number of young is between $6.9$ and $8.3$

Part (c) Because there is only $95\%$ confidence that in all-female eastern cottonmouth litters, the average number of young is between $6.9$ and $8.3.$

Part (a) Step 1: Given information

$\sigma =2.4$

Part (a) Step 2: Concept

The formula used: $\overline{x}\pm 2\frac{\sigma }{\sqrt{n}}$

Part (a) Step 3: Calculation

For all female eastern cottonmouths, calculate a $95$ percent confidence interval for the mean number of young each litter.

From Exercise, $8-18E$, $\overline{x}=7.6, n=44$, and consider $\sigma =2.4$

Empirical rule:

Property 1: Around $68\%$ of the data set is located between $(\overline{x}-s,\overline{ x}+s)$

Property 2: Approximately $95\%$ of the data set is located between $(\overline{x}-2s,\overline{ x}+2s)$

Property 3: Approximately $99.7\%$ of the data set is located between $(\overline{x}-3s,\overline{ x}+3s)$

Using Property $2$, $95\%$ of all observations fall within two standard deviations of the mean on either side.

The $95\%$ confidence interval for the population mean is,

$$\begin{gathered} \overline{x}\pm 2\frac{\sigma }{\sqrt{n}}=7.6\pm 2\left(\frac{2.4}{\sqrt{44}}\right) \\ =7.6\pm 0.7 \\ =(7.6-0.7,7.6-0.7) \\ =(6.9,8.3) \end{gathered}$$

As a result, a $95$ percent confidence interval was established for the mean number of young per litter of all-female eastern cottonmouths $(6.9,8.3)$

Part (b) Step 1: Explanation

Eastern cottonmouths with all females have an average of $6.9$ to $8.3$young per litter. According to $95\%$ confidence.

Part (c) Step 1: Explanation

Because there is only $95\%$ confidence that the mean number of a young per litter of all-female eastern cottonmouths is between $6.9$ and $8.3$ the mean number of a young per litter of all-female eastern cottonmouths may or may not lie in the confidence interval obtained in part (a).