Question

For a normal distribution, a positive value of z indicates that _______.
a. all the observations must have had positive values.
b. the area corresponding to the z is either positive or negative.
c. the sample mean is smaller than the population mean.

Answer 1

Answer:

A positive value of z indicates the sample mean is more than the population mean.

Step-by-step explanation:

The z-score or the standardized normal distribution score is defined as:

                                            $z=\frac{\bar x-\mu}{{\frac{\sqrt{\sigma} }{n} } }$

Here,

$\bar x$ = sample mean

$\mu$ = population mean

$\sigma$ = population standard deviation

n = sample size.

The value of population standard deviation is always positive since it is the square of the deviations of values from their mean.

Then if the value of z is positive or negative is determined by the value of $\bar x$ and $\mu$.

• If z = positive, this implies that

          $\bar x-\mu>0\\\bar x>\mu$

       That is the sample mean is greater than the population mean.

• If z = negative, this implies that

          $\bar x-\mu<0\\\bar x<\mu$

       That is the sample mean is smaller than the population mean.

Thus, a positive value of z indicates the sample mean is more than the population mean.