Problem
Calculate the standard normal z-score for a sample mean of 18.8: A random sample of $n=64$ observations is drawn from a population with a mean equal to 19 and a standard deviation equal to 16 . Complete parts a through $g$ below. c. Calculate the standard normal $z$-score corresponding to a value of $\bar{x}=18.8$. $z=-0.1$ (Type an integer or a decimal.) d. Calculate the standard normal $z$-score corresponding to a value of $\bar{x}=22.4$. $z=1.7$ (Type an integer or a decimal.) e. Find $P(\bar{x}<18.8)$. $P(\bar{x}<18.8)=0.4602$ (Round to four decimal places as needed.) f. Find $P(\bar{x}>22.4)$. $P(\bar{x}>22.4)=0.0446$ (Round to four decimal places as needed.) g. Find $P(18.8<\bar{x}<22.4)$. $P(18.8<\bar{x}<22.4)=\square$ (Round to four decimal places as needed.)
Solution
Step 1 ：Given that the sample mean (\(\bar{x}\)) is 18.8, the population mean (\(\mu\)) is 19, the standard deviation (\(\sigma\)) is 16, and the number of observations (\(n\)) is 64.
Step 2 ：The formula for the z-score is: \[z = \frac{\bar{x} - \mu}{\sigma / \sqrt{n}}\]
Step 3 ：Substitute the given values into the formula: \[z = \frac{18.8 - 19}{16 / \sqrt{64}}\]
Step 4 ：Simplify the expression to find the z-score: \[z = -0.1\]
Step 5 ：Final Answer: The standard normal z-score corresponding to a sample mean of 18.8 is \(\boxed{-0.1}\).
