Te weight of an adult bottlenose dolphin was found to follow a normal distribution with a mean of 550 pounds and a standard deviation of 50 pounds. a. What percentage 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 X (500 580 < < ) ? c. In a random sample of 9 adult bottlenose dolphins, what is the probability that 5 of them are heavier than 560 pounds?

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 = 0.0013at x = 600[tex]Z=\frac{600-550}{50}[/tex][tex]Z=1[/tex]Refer the z table for p value p value = 0.8413P(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 poundsb)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 = 0.1587at x = 580[tex]Z=\frac{580-550}{50}[/tex][tex]Z=0.6[/tex]Refer the z table for p value p value = 0.7257P(500<x<580)=P(x<580)-P(x<500)=0.7257-0.1587=0.84c). 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 = 0.5793P(x>560)=1-P(x<560)=1-0.5793=0.4207Now 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 0.4207q = 1-p = probability of failuren = 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