Question

On the distant planet Cowabunga, the weights of cows have a normal distribution with a mean of 479 pounds and a standard deviation of 40 pounds. The cow transport truck holds 15 cows and can hold a maximum weight of 7350. If 15 cows are randomly selected from the very large herd to go on the truck, what is the probability their total weight will be over the maximum allowed of 7350

Answer 1

Answer: 0.1435

Step-by-step explanation:

Given : Mean = 479 pounds

Standard deviation = 40 pounds.

Let  X denote the weights of cows.

$X\sim N(\mu=479,\sigma=40)$

The cow transport truck holds 15 cows and can hold a maximum weight of 7350.

i.e. mean weight of cow in this case =$\overline{x}=\dfrac{7350}{15}=490\text{ pounds}$

If 15 cows are randomly selected from the very large herd to go on the truck, what is the probability their total weight will be over the maximum allowed of 7350 will be:-

$P(\overline{x}>490)=P(\dfrac{\overline{x}-\mu}{\dfrac{\sigma}{\sqrt{n}}}>\dfrac{490-479}{\dfrac{40}{\sqrt{15}}})\\\\=P(z>1.065)\ \ [\because\ z=\dfrac{\overline{x}-\mu}{\dfrac{\sigma}{\sqrt{n}}}]\\\\=1-P(z\leq1.065)\\\\=1- 0.8565=0.1435\ \ [\text{By z-table}]$

Hence, If 15 cows are randomly selected from the very large herd to go on the truck,  the probability their total weight will be over the maximum allowed of 7350 = 0.1435