Q. 3.195 - Elementary Statistics

Question

Fill in the following blanks.

(a) A standardized variable always has mean _____ and standard deviation _______ .

(b) The z-score corresponding to an observed value of a variable tells you ______ .

(c) A positive z-score indicates that the observation is ______ the mean, whereas a negative z-score indicates that the observation is _______ the mean.

Step-by-Step Solution

Answer

(a) A standardized variable always has a mean of $0$ and a standard deviation of $1$.

(b) The $Z$-score corresponding to an observed value of a variable tells you standard normal score 

(c) A positive $Z$-score indicates that the observation is  to right of  the mean, whereas a negative z-score indicates that the observation is   to the left of  the mean  

Part(a) Step 1: Given information

We need to fill up blanks.

Part(a) Step 2: Explanation

A standardized variable for normal variable X is

$Z=(x-\mu )/\sigma$

 will be normal with mean =$0$ and sigma = $1$ 

Hence the answers would be: A standardized variable always has a mean of $\underline{0}$ and a standard deviation of $\underline{1}$.

Part(b) step 1: Given information

We need to fill in blanks.

Part(b) Step 2: Explanation

In statistics, $Z$ -score is a numerical measurement of a value's relationship to the mean (average) of a collection of values, expressed in standard deviations from the mean.

$Z = (x-\mu )/\sigma$

Hence answer would be, The $Z$-score corresponding to an observed value of a variable tells you standard normal score

Part(c) Step 1: Given information

We need to fill in blanks.

Part(c) Step 2: Explanation

We know formula of $Z$-score is $Z=(x-\mu )/\sigma$

Hence answer is 

A positive $Z$-score indicates that the observation is  to right of  the mean, whereas a negative $Z$-score indicates that the observation is   to the left of  the mean