Question

HELP PLEASE WITH WORK SHOWN

A distribution of values is normal with a mean of 170 and a standard deviation of 12. From this distribution, you are dra wing samples of size 37. Find the interval containing the middle-most 50% of sample means Enter your answer using interval notation. In this context, either inclusive or exclusive intervals would be acceptable. Your numbers should be accurate to 1 decimal places. Answers obtained using exact z-scores or z-scores rounded to 3 decimal places are accepted.

Answer 1

Let X be the normal random variable.

Given, $X \sim N(170,12^{2})$

i.e. $\mu_{x} = 170, \sigma_{x} = 12$

Sample size, n = 37

Now, let us denote the sample mean by $\bar{X}$

Now, corresponding interval for Z-score for the middle-most 50% of sample means is (-0.674, 0.674)

Now, $\bar{X} \sim N(\mu_{X}, \frac{\sigma_{X}^{2}}{n})$

Thus, The interval is { (170 - 0.674*12/√37), (170 + 0.674*12/√37) }

= ( 168.7, 171.3 )