Q.3.214 - Elementary Statistics

Question

 Gestation Periods of Humans. Gestation periods of humans have a mean of $266$days and a standard deviation of $16$days. Let $y$ denote the variable "gestation period" for humans.

a. Find the standardized variable corresponding to $y$

b. What are the mean and standard deviation of the standardized variable?

c. Obtain the $z$-scores for gestation periods of $227$ days and $315$days. Round your answers to two decimal places.

d. Interpret your answers in part (c).

c. Construct a graph similar to Fig. 3.20 on page 146 that shows your results from parts (b) and (c).

Step-by-Step Solution

Answer

(a) The standardized version is$\frac{y-266}{16}$.

(b) Mean$=0$and Standard deviation $=1$

(c) The $z$- scores are $-2.44\&3.06$

(d) The first prisoner's$z$ score revealed a standard deviation of $2.44$ below the mean for prison time served.

The second prisoner's $z$ score revealed a standard deviation of $3.06$ from the mean for prison time served.

(e) 

Part (a) Step 1: Given Information

Given in the question that the gestation periods of humans have a mean of $266$days and a standard deviation of $16$ days. Let $y$denote the variable "gestation period" for humans. we have to find the standardized variable corresponding to  $y$.

Part(a) Step 2: Explanation

$y\text{'}s$ standardized variable is determined as follows:

$$\begin{gathered} z=\frac{y-\mu }{\sigma } \\ =\frac{y-266}{16} \end{gathered}$$

Part (b) Step 1: Given Information

Given in the question that the gestation periods of humans have a mean of$266$days and a standard deviation of  $16$days. Let  $y$denote the variable "gestation period" for humans. we have to find the mean and standard deviation of the standardized variable .

Part(b) Step 2: Explanation

The mean is computed as follows

$$\begin{gathered} {\mu }_{\mathrm{z}}=\frac{\sum \frac{y_{\mathrm{i}}-\mu }{\sigma }}{n} \\ =\frac{\sum y_i-n\mu }{\sigma n} \\ =\frac{n\mu -n\mu }{\sigma n} \\ =0 \end{gathered}$$

The standard deviation  is computed as follows 

$$\begin{gathered} {\sigma }_{\mathrm{Z}}=\sqrt{\frac{\sum {\left(z_i-{\mu }_z\right)}^2}{n}} \\ =\frac{\sigma }{\sigma } \\ =1 \end{gathered}$$

Part (c) Step 1: Given Information

Given in the question that the gestation periods of humans have a mean of $266$days and a standard deviation of  $16$days. Let  $y$denote the variable "gestation period" for humans. we have to find the $z$-scores for gestation periods of $227$days and $315$days.

Part(c) Step 2: Explanation

score is calculated as

$$\begin{gathered} z_1=\frac{y_1-\mu }{\sigma } \\ =\frac{227-266}{16} \\ \approx -2.44 \end{gathered}$$

$$\begin{gathered} z_2=\frac{y_2-\mu }{\sigma } \\ =\frac{315-266}{16} \\ =3.06 \end{gathered}$$

Part (d) Step 1: Given Information

Given in the question that the gestation periods of humans have a mean of $266$days and a standard deviation of $16$days. Let $y$denote the variable "gestation period" for humans. we have to Interpret your answers in part (c)  

Part(d) Step 2: Explanation

The first prisoner's $z$score revealed a standard deviation of $2.44$below the mean for prison time served.

The second prisoner's $z$ score revealed a standard deviation of $3.06$ from the mean for prison time served. 

Part (e) Step 1: Given Information

Given in the question that the gestation periods of humans have a mean of  $266$days and a standard deviation of  $16$days. Let$y$denote the variable "gestation period" for humans. we have to construct a graph.

Part(e) Step 2: Explanation