Question

the amount of time to complete a physical activity in a pe class is approximately normally distributed with a mean of 38 seconds and a standard deviation of 5.5 seconds. a) what is the probability that a randomly chosen student completes the activity in less than 32.6 seconds? (round answer to three decimal places) b) what is the probability that a randomly chosen student completes the activity in more than 44.7 seconds? (round answer to three decimal places) c) what proportion of students take between 35.6 and 42.8 seconds to complete the activity? (round answer to three decimal places) d) 80% of all students finish the activity in less than seconds. (round answer to one decimal places)

Explanation:

Step1: Calculate z - score formula

The z - score formula is $z=\frac{x-\mu}{\sigma}$, where $x$ is the value from the data set, $\mu$ is the mean, and $\sigma$ is the standard deviation. Given $\mu = 38$ and $\sigma=5.5$.

Step2: Solve part a

For $x = 32.6$, $z=\frac{32.6 - 38}{5.5}=\frac{- 5.4}{5.5}\approx - 0.982$. Using the standard normal table, $P(X\lt32.6)=P(Z\lt - 0.982)\approx0.163$.

Step3: Solve part b

For $x = 44.7$, $z=\frac{44.7-38}{5.5}=\frac{6.7}{5.5}\approx1.218$. Then $P(X\gt44.7)=1 - P(X\leq44.7)=1 - P(Z\leq1.218)$. From the standard - normal table, $P(Z\leq1.218)\approx0.889$, so $P(X\gt44.7)=1 - 0.889 = 0.111$.

Step4: Solve part c

For $x_1 = 35.6$, $z_1=\frac{35.6 - 38}{5.5}=\frac{-2.4}{5.5}\approx - 0.436$. For $x_2 = 42.8$, $z_2=\frac{42.8 - 38}{5.5}=\frac{4.8}{5.5}\approx0.873$. Then $P(35.6\lt X\lt42.8)=P(-0.436\lt Z\lt0.873)=P(Z\lt0.873)-P(Z\lt - 0.436)$. From the standard - normal table, $P(Z\lt0.873)\approx0.809$ and $P(Z\lt - 0.436)\approx0.331$. So $P(35.6\lt X\lt42.8)=0.809 - 0.331 = 0.478$.

Step5: Solve part d

We want to find $x$ such that $P(X\lt x)=0.80$. Looking up the z - score in the standard - normal table for a probability of $0.80$, the z - score $z\approx0.842$. Then, using the z - score formula $z=\frac{x-\mu}{\sigma}$, we have $0.842=\frac{x - 38}{5.5}$. Solving for $x$: $x=38+0.842\times5.5=38 + 4.631\approx42.6$.

Answer:

a) $0.163$
b) $0.111$
c) $0.478$
d) $42.6$