Find $\mathrm { P } ( \bar { X }

Question

Find $\mathrm { P } ( \bar { X } < 25 )$ If ? = 16 and $\sigma _ { \bar { X } } = 4$
A)1.000
B)2.25
C).9878
D).0122

Quizplus · Accepted Answer

To begin, we need to standardize the variable of interest, which is the sample mean. Since we know that the population mean is 16 and the standard deviation of the sample mean is 4, we can use the formula:

$$Z = \frac{\bar{X} - \mu}{\sigma_{\bar{X}}} = \frac{25-16}{4} = 2.25$$

where Z is the standard normal variable.

Next, we want to find the probability that the sample mean is less than 25, which is equivalent to finding the probability that Z is less than:

$$Z = \frac{25-16}{4} = 2.25$$

Using a standard normal table or calculator, we can find that the area to the left of 2.25 is approximately 0.9878.

Finally, we want the probability that the sample mean is greater than 25, which is the complement of the probability that the sample mean is less than or equal to 25. Therefore:

$$\mathrm{P}(\bar{X} < 25) = 0.9878$$

So the answer is C.

Find P(Xˉ<25)\mathrm { P } ( \bar { X } < 25 )P(Xˉ<25)

Find $\mathrm { P } ( \bar { X } < 25 )$