Question

a sample of size 86 will be drawn from a population with mean 90 and standard deviation 24. use excel.
part: 0 / 2
part 1 of 2
(a) find the probability that \\( \overline{x} \\) will be less than 89. round the answer to at least four decimal places.
the probability that \\( \overline{x} \\) will be less than 89 is 0.3507.

Explanation:

Step1: Calculate the standard error

The standard error (SE) is calculated as $\frac{\sigma}{\sqrt{n}}$, where $\sigma = 24$ and $n = 86$. So, $SE=\frac{24}{\sqrt{86}}\approx\frac{24}{9.2736}\approx2.588$.

Step2: Calculate the z - score

The z - score is calculated as $z=\frac{\bar{x}-\mu}{SE}$, where $\bar{x} = 89$, $\mu = 90$ and $SE\approx2.588$. So, $z=\frac{89 - 90}{2.588}\approx\frac{- 1}{2.588}\approx - 0.386$.

Step3: Use Excel to find the probability

In Excel, we can use the NORM.DIST function. The formula is NORM.DIST(x, mean, standard_dev, cumulative). Here, $x = 89$, mean $=\mu=90$, standard_dev $=SE\approx2.588$, and cumulative $=TRUE$.
Using Excel: NORM.DIST(89,90,24/SQRT(86),TRUE)
First, calculate $24/\sqrt{86}\approx2.588$. Then, NORM.DIST(89,90,2.588,TRUE) gives the probability.
Calculating this, we get approximately 0.3497.

Answer:

0.3497 (rounded to four decimal places)