Suppose you take a random sample from a normally distributed population with mean $\mu$
= 16 and compute the sample mean $\bar { x }$
.If $\sigma _ { \bar { x } } ^ { 2 }$
= 4,find P( $\bar { x }$

Question

Suppose you take a random sample from a normally distributed population with mean $\mu$
= 16 and compute the sample mean $\bar { x }$
.If $\sigma _ { \bar { x } } ^ { 2 }$
= 4,find P( $\bar { x }$
< 25).
A)0300
B)1.000
C)1732
D)0577
E)8413

Quizplus · Accepted Answer

Given that the variance of the sample mean, $\sigma _ { \bar { x } } ^ { 2 }$ = 4, the standard deviation of the sample mean, $\sigma _ { \bar { x } }$, is the square root of 4, which is 2. To find P($\bar { x }$ < 25), we calculate the Z-score using the formula Z = ($\bar { x }$ - $\mu$) / $\sigma _ { \bar { x } }$. Substituting $\bar { x }$ = 25, $\mu$ = 16, and $\sigma _ { \bar { x } }$ = 2 gives Z = (25 - 16) / 2 = 4.5. The Z-score of 4.5 corresponds to a probability very close to 1.000, indicating that the probability of the sample mean being less than 25 is almost certain.

Suppose You Take a Random Sample from a Normally Distributed μ\muμ

Suppose You Take a Random Sample from a Normally Distributed $\mu$