a certain standardized tests math scores have a bell - shaped distribution with a mean of 520 and a standard…
a certain standardized tests math scores have a bell - shaped distribution with a mean of 520 and a standard deviation of 119. complete parts (a) through (c). (a) what percentage of standardized test scores is between 401 and 639? 68.3 % (round to one decimal place as needed.) (b) what percentage of standardized test scores is less than 401 or greater than 639? % (round to one decimal place as needed.)
Answer
Explanation:
Step1: Calculate z - scores
The z - score formula is $z=\frac{x-\mu}{\sigma}$, where $\mu = 520$ is the mean and $\sigma = 110$ is the standard deviation. For $x = 401$, $z_1=\frac{401 - 520}{110}=\frac{- 119}{110}\approx - 1.08$. For $x = 639$, $z_2=\frac{639 - 520}{110}=\frac{119}{110}\approx1.08$.
Step2: Find probabilities for part (a)
We want to find $P(401<X<639)$. Using the standard normal distribution table, $P(-1.08 < Z < 1.08)=P(Z < 1.08)-P(Z < - 1.08)$. From the standard - normal table, $P(Z < 1.08)=0.8599$ and $P(Z < - 1.08)=0.1401$. So $P(-1.08 < Z < 1.08)=0.8599 - 0.1401 = 0.7198\approx72.0%$.
Step3: Find probabilities for part (b)
We want to find $P(X<401\ or\ X > 639)$. Since $P(401<X<639)=0.7198$, then $P(X<401\ or\ X > 639)=1 - P(401<X<639)$. So $P(X<401\ or\ X > 639)=1 - 0.7198 = 0.2802\approx28.0%$.
Answer:
(a) $72.0%$ (b) $28.0%$