Question

the fictional college aptitude test (cat) has scores that are normally distributed with a mean of 1156 and a standard deviation of 223 points. a. what is the probability that a randomly selected student taking the cat scores above 952? you may round your answer to four decimals. 180 × 0.8199 b. find $p_{46}$, the 46th percentile for the cat scores - the score that separates the lower 46% of the test scores from the rest. round to the nearest whole number. c. if a random sample of 43 students is given the cat, what is the probability that the sample mean of their test scores is below 1219? round to four decimal places.

Explanation:

Step1: Calculate z - score for part a

The z - score formula is $z=\frac{x-\mu}{\sigma}$, where $\mu = 1156$, $\sigma=223$, and $x = 952$. So, $z=\frac{952 - 1156}{223}=\frac{- 204}{223}\approx - 0.9148$.

Step2: Find probability for part a

We want $P(X>952)$, which is equivalent to $P(Z>-0.9148)$. Since $P(Z > - 0.9148)=1 - P(Z\leq - 0.9148)$, and from the standard normal table $P(Z\leq - 0.9148)\approx0.1801$, so $P(Z>-0.9148)=1 - 0.1801 = 0.8199$.

Step3: Find z - score for part b

We want to find the z - score $z$ such that $P(Z\leq z)=0.46$. Looking up in the standard - normal table, the z - score corresponding to a probability of $0.46$ is approximately $z=-0.10$.

Step4: Calculate the score for part b

Using the formula $x=\mu+z\sigma$, with $\mu = 1156$, $\sigma = 223$, and $z=-0.10$, we get $x=1156+( - 0.10)\times223=1156 - 22.3\approx1134$.

Step5: Calculate z - score for part c

The standard deviation of the sample mean (standard error) is $\sigma_{\bar{x}}=\frac{\sigma}{\sqrt{n}}$, where $\sigma = 223$, $n = 43$. So, $\sigma_{\bar{x}}=\frac{223}{\sqrt{43}}\approx33.97$. The z - score for $\bar{x}=1219$ is $z=\frac{\bar{x}-\mu}{\sigma_{\bar{x}}}=\frac{1219 - 1156}{33.97}=\frac{63}{33.97}\approx1.85$.

Step6: Find probability for part c

We want $P(\bar{X}<1219)$, which is equivalent to $P(Z < 1.85)$. From the standard - normal table, $P(Z < 1.85)=0.9678$.

Answer:

a. $0.8199$
b. $1134$
c. $0.9678$