Q 8.23. - Elementary Statistics

Question

Giant Tarantulas. A tarantula has two body parts. The anterior part of the body is covered above by a shell. or carapace. In the paper "Reproductive Biology of Uruguayan Theraphosids" (The Journal of Arachnology, Vol. $30$, No. $3$ Pp. $571-587$), F. Costa and F. Perez-Miles discussed a large species of tarantula whose common name is the Brazilian giant tawny red. A simple random sample of $15$ of these adult male tarantulas provided the following data on carapace length, in millimeters $\left(\mathrm{mm}\right)$ 

a. Obtain a normal probability plot of the data.

b. Based on your result from part (a), is it reasonable to presume that the carapace length of adult male Brazilian giant tawny red tarantulas is normally distributed? Explain your answer.

c. Find and interpret a $95\%$ confidence interval for the mean carapace length of all adult male Brazilian giant tawny red tarantulas. The population standard deviation is $1.76 \mathrm{mm}$

d. In Exercise $6.97$ we noted that the mean carapace length of all adult male Brazilian giant tawny red tarantulas is \(18.14 \mathrm{~mm}\). Does your confidence interval in part (c) contain the population means? Would it necessarily have to? Explain your answers.

Step-by-Step Solution

Answer

Part (a) It can be concluded that the distribution of length is an approximately normal distribution.

Part (b) Yes.

Part (c) The $95$ percent confidence interval for all adult male Brazilian enormous tawny red tarantulas' mean carapace length is$(17.52,19.34)$

Part (d) Yes, the population means is included in the confidence interval in section (c). This happens around $5\%$ of the time.

Part (a) Step 1: Given information

15.7	18.3	19.7	17.6	19.0
19.2	19.8	18.1	18.0	20.9
16.4	16.8	18.9	18.5	19.5

Part (a) Step 2: Explanation

MINITAB is used to create the normal probability plot.

Procedure for MINITAB:

Step 1: Select Probability Plot from the Graph menu.

Step 2: Click OK after selecting Single.

Step 3: In the Graph variables section, add the Length column.

Step 4: Click the OK button. 

MINITAB OUTPUT: 

All observations are closer to a straight line on the probability plot of length. As a result, it may be argued that the length distribution is approximately normal.

Part (b) Step 1: Explanation

Because the data are quite close to a straight line, it is plausible to assume that the carapace length of mature male Brazilian huge tawny red tarantulas is regularly distributed.

Part (c) Step 1: Calculation

Find the $95\%$ confidence interval for all adult male Brazilian giant tawny red tarantulas' mean carapace length.

The sample means is$18.43 \mathrm{mm}$according to a normal probability plot.

The population standard deviation is $1.76 \mathrm{mm}$ according to the data.

Empirical rule:

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

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

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

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

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

$$\begin{gathered} \overline{x}\pm 2\frac{\sigma }{\sqrt{n}}=18.43\pm 2\left(\frac{1.76}{\sqrt{15}}\right) \\ =18.43\pm 0.91 \\ =(18.43-0.91,18.43+0.91) \\ =(17.52,19.34) \end{gathered}$$

The $95\%$ confidence interval for the mean carapace length of all adult male Brazilian huge tawny red tarantulas is thus $(17.52,19.34)$

Part (d) Step 1: Explanation

The mean carapace length of all mature male Brazilian huge tawny red tarantulas is $18.14 \mathrm{mm}$ according to Exercise $6.7$ This price ranges from $17.52$ to $19.34$

Yes, the population mean is included in the confidence interval in section (c). This happens around$5\%$ of the time.