Respuesta :
Answer:
a) [tex]P(\bar X >138)=P(Z>\frac{138-179.4}{\frac{37.7}{\sqrt{70}}}=-9.187)[/tex]
And using the complement rule we got:
[tex] P(Z>-9.187) = 1-P(Z<-0.987)=1-0.0001 =0.9999[/tex]
b) [tex]P(\bar X >173)=P(Z>\frac{173-179.4}{\frac{37.7}{\sqrt{17}}}=-0.700)[/tex]
And using the complement rule we got:
[tex] P(Z>-0.700) = 1-P(Z<-0.700)=1-0.2420 =0.7580[/tex]
c) B. Because the probability of overloading is lower with the new ratings than with the old ratings, the new ratings appear to be safe.
Step-by-step explanation:
Previous concepts
Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".
The Z-score is "a numerical measurement used in statistics of a value's relationship to the mean (average) of a group of values, measured in terms of standard deviations from the mean".
Part a
Let X the random variable that represent the weights of a population, and for this case we know the distribution for X is given by:
[tex]X \sim N(179.4,37.7)[/tex]
Where [tex]\mu=179.4[/tex] and [tex]\sigma=37.7[/tex]
For this case since the distirbution for X is normal then the distribution for the sample mean [tex]\bar X [/tex] is also normal and given by:
[tex]\bar X \sim N(\mu, \frac{\sigma}{\sqrt{n}})[/tex]
If we apply this formula we got:
[tex]P(\bar X >138)=P(Z>\frac{138-179.4}{\frac{37.7}{\sqrt{70}}}=-9.187)[/tex]
And using the complement rule we got:
[tex] P(Z>-9.187) = 1-P(Z<-0.987)=1-0.0001 =0.9999[/tex]
Part b
[tex]P(\bar X >173)=P(Z>\frac{173-179.4}{\frac{37.7}{\sqrt{17}}}=-0.700)[/tex]
And using the complement rule we got:
[tex] P(Z>-0.700) = 1-P(Z<-0.700)=1-0.2420 =0.7580[/tex]
Part c
B. Because the probability of overloading is lower with the new ratings than with the old ratings, the new ratings appear to be safe.
The probabilities are as follows
A. The probability that the boat is overloaded because the 70 passengers have a mean weight greater than 138 lb is 0.9999.
B. The probability that the boat is overloaded because the mean weight of the passengers is greater than 173 is 0.7580.
C. The correct option is B.
What is normal a distribution?
It is also called the Gaussian Distribution. It is the most important continuous probability distribution. The curve looks like a bell, so it is also called a bell curve.
A boat capsized and sank in a lake.
Based on an assumption of mean weight of 138 lb, the boat was rated to carry 70 passengers (so the load limit was 9,660 lb).
After the boat sank, the assumed mean weight for similar boats was changed from 138 lb to 173 lb.
The z-score is a numerical measurement used in statistics of the value's relationship to the mean of a group of values, measured in terms of standards from the mean.
A. Let X be the random variable that represents the weights of a population for this case we know that the distribution for X is given as
[tex]\rm X \sim N (179.4, 37.7)[/tex]
where
[tex]\mu = 179.4\\\sigma = 37.7[/tex]
For this case since the distribution for X is normal then the distribution for the sample mean [tex]\rm \bar{X}[/tex] is also normal and given as
[tex]\rm \bar{X} \sim N(\mu , \dfrac{\sigma}{\sqrt{n}})[/tex]
Then
[tex]\rm P(\bar{X} > 138 ) = P(z > \dfrac{138 - 179 .4}{\frac{37.7}{\sqrt{70}}} = -9.187)[/tex]
And using the complement rule, we get
[tex]\rm P(Z > - 9.187) = 1-P(Z < -9.187) = 1 - 0.0001 = 0.9999[/tex]
B.
[tex]\rm P(\bar{X} > 173) = P(z > \dfrac{173- 179 .4}{\frac{37.7}{\sqrt{70}}} = -0.700)[/tex]
And using the complement rule, we get
[tex]\rm P(Z > - 0.700) = 1-P(Z < -0.700) = 1 - 0.2420= 0.7580[/tex]
C. The probability of overloading is lower with the new rating than with the old ratings, the new ratings appear to be safe.
More about the normal distribution link is given below.
https://brainly.com/question/12421652
