stats lesson 2.5 homework name emmalynne here. 1. for a standardized normal distribution, find the area…
stats lesson 2.5 homework name emmalynne here. 1. for a standardized normal distribution, find the area described by each of the following inequalities: a. z < - 0.80 b. - 1.93 < z < 1.38 c. z > 2.26 d. z ≥ 2.26 2. researchers measured the resting heart rate of over 92,000 individuals using wearable devices over a two - year period. the results of the study showed the resting heart rates were approximately normal with a mean of 63 beats per minute (bpm) and a standard deviation of 3.03 bpm. a. what proportion of the individuals in the study have a resting heart rate below 58 bbm? b. what proportion of the individuals in the study have a resting heart rate above 66 bbm? c. what proportion of the individuals in the study have a resting heart rate between 58 and 66 bbm? d. athletes often have lower than average resting heart rates. what is the resting heart rate for an athlete who is at the 10th percentile.
Answer
1.
a.
Explanation:
Step1: Use z - table lookup
We know that for a standard - normal distribution, we want to find $P(Z < - 0.80)$. Looking up the value in the standard - normal (z -) table, the value corresponding to $z=-0.80$ is $0.2119$.
Answer:
$0.2119$
b.
Explanation:
Step1: Split the inequality
We know that $P(-1.93 < Z < 1.38)=P(Z < 1.38)-P(Z < - 1.93)$.
Step2: Look up values in z - table
From the z - table, $P(Z < 1.38)=0.9106$ and $P(Z < - 1.93)=0.0268$.
Step3: Calculate the probability
$P(-1.93 < Z < 1.38)=0.9106 - 0.0268=0.8838$.
Answer:
$0.8838$
c.
Explanation:
Step1: Use the property of the total area
We know that $P(Z>2.26)=1 - P(Z < 2.26)$.
Step2: Look up the value in z - table
From the z - table, $P(Z < 2.26)=0.9881$.
Step3: Calculate the probability
$P(Z>2.26)=1 - 0.9881 = 0.0119$.
Answer:
$0.0119$
d.
Explanation:
Step1: Use the property of the total area
Since the standard - normal distribution is continuous, $P(Z\geq2.26)=P(Z > 2.26)=1 - P(Z < 2.26)$.
Step2: Look up the value in z - table
From the z - table, $P(Z < 2.26)=0.9881$.
Step3: Calculate the probability
$P(Z\geq2.26)=1 - 0.9881=0.0119$.
Answer:
$0.0119$
2.
Let $\mu = 63$ and $\sigma=3.03$. The z - score is calculated as $z=\frac{x-\mu}{\sigma}$.
a.
Explanation:
Step1: Calculate the z - score
For $x = 58$, $z=\frac{58 - 63}{3.03}=\frac{-5}{3.03}\approx - 1.65$.
Step2: Look up the value in z - table
Looking up $z=-1.65$ in the z - table, $P(Z < - 1.65)=0.0495$.
Answer:
$0.0495$
b.
Explanation:
Step1: Calculate the z - score
For $x = 66$, $z=\frac{66 - 63}{3.03}=\frac{3}{3.03}\approx0.99$.
Step2: Use the property of the total area
$P(Z>0.99)=1 - P(Z < 0.99)$.
Step3: Look up the value in z - table
From the z - table, $P(Z < 0.99)=0.8389$.
Step4: Calculate the probability
$P(Z>0.99)=1 - 0.8389 = 0.1611$.
Answer:
$0.1611$
c.
Explanation:
Step1: Calculate the z - scores
For $x = 58$, $z_1=\frac{58 - 63}{3.03}\approx - 1.65$. For $x = 66$, $z_2=\frac{66 - 63}{3.03}\approx0.99$.
Step2: Calculate the probability
$P(-1.65 < Z < 0.99)=P(Z < 0.99)-P(Z < - 1.65)$.
Step3: Look up the values in z - table
$P(Z < 0.99)=0.8389$ and $P(Z < - 1.65)=0.0495$.
Step4: Calculate the probability
$P(-1.65 < Z < 0.99)=0.8389 - 0.0495=0.7894$.
Answer:
$0.7894$
d.
Explanation:
Step1: Find the z - score for the 10th percentile
Looking up the 10th percentile (or 0.1000) in the body of the z - table, the closest z - score is approximately $z=-1.28$.
Step2: Use the z - score formula to find x
We know that $z=\frac{x-\mu}{\sigma}$, so $x=\mu+z\sigma$.
Step3: Substitute the values
Substituting $\mu = 63$, $z=-1.28$, and $\sigma = 3.03$, we get $x=63+( - 1.28)\times3.03=63-3.88=59.12$.
Answer:
$59.12$