Respuesta :
Answer:
Mean = [tex]\mu = 550[/tex]
Standard deviation = [tex]\sigma = 50[/tex]
a. What percentage of adult bottlenose dolphins weigh from 400 to 600 pounds?
P(400<x<600)
Formula : [tex]Z=\frac{x-\mu}{\sigma}[/tex]
at x = 400
[tex]Z=\frac{400-550}{50}[/tex]
[tex]Z=-3[/tex]
Refer the z table for p value
p value = 0.0013
at x = 600
[tex]Z=\frac{600-550}{50}[/tex]
[tex]Z=1[/tex]
Refer the z table for p value
p value = 0.8413
P(400<x<600)=P(x<600)-P(x<400)=0.8413-0.0013=0.84
So,84% of adult bottlenose dolphins weigh from 400 to 600 pounds
b)If X represents the mean weight of a random sample of 9 adult bottlenose dolphins, what is P (500<x < 580) ?
Formula : [tex]Z=\frac{x-\mu}{\sigma}[/tex]
at x = 500
[tex]Z=\frac{500-550}{50}[/tex]
[tex]Z=-1[/tex]
Refer the z table for p value
p value = 0.1587
at x = 580
[tex]Z=\frac{580-550}{50}[/tex]
[tex]Z=0.6[/tex]
Refer the z table for p value
p value = 0.7257
P(500<x<580)=P(x<580)-P(x<500)=0.7257-0.1587=0.84
c). In a random sample of 9 adult bottlenose dolphins, what is the probability that 5 of them are heavier than 560 pounds?
at x = 560
[tex]Z=\frac{560-550}{50}[/tex]
[tex]Z=0.2[/tex]
Refer the z table for p value
p value = 0.5793
P(x>560)=1-P(x<560)=1-0.5793=0.4207
Now to find the the probability that 5 of them are heavier than 560 pounds we will use binomial distribution
[tex]P(X=r)=^nC_r p^r q^{n-r}[/tex]
p is the probability of success that is 0.4207
q = 1-p = probability of failure
n = 9
r = 5
[tex]P(X=5)=^9C_5 (0.4207)^5 (1-0.4207)^{9-5}[/tex]
[tex]P(X=5)=\frac{9!}{5!(9-5)!}(0.4207)^5 (1-0.4207)^{9-5}[/tex]
[tex]P(X=5)=0.187[/tex]
Hence In a random sample of 9 adult bottlenose dolphins, the probability that 5 of them are heavier than 560 pounds is 0.187
