Respuesta :
Answer:
a) The probability that a hobbit picked at random is no more than 36.7 in tall is P= 0.56631.
b) The probability that a random sample of 16 hobbits have a mean height of no more than 36.7 in tall is P=0.74857.
c) The probability that a random sample of 100 hobbits have a mean height of no more than 36.7 in tall is P=0.95254.
Step-by-step explanation:
We have this parameters for the population: normally distributed with mean 36 in. and standard deviation 4.2 in.
a) The probability can be computed calculating z and looking up in a table.
[tex]z=\frac{x-\mu}{\sigma}=\frac{36.7-36}{4.2}=\frac{0.7}{4.2}=0.167[/tex]
The probability that a hobbit picked at random is no more than 36.7 in tall is P= 0.56631.
[tex]P(x\leq36.7)=P(z\leq 0.167)=0.56631[/tex]
b) In this case, the sample standard deviation change to
[tex]\sigma_s=\frac{\sigma}{\sqrt{n} }[/tex].
We can calculate z with the sample standard deviation
[tex]z=\frac{x-\mu}{\sigma/\sqrt{n}}=\frac{36.7-36}{4.2/\sqrt{16}}=\frac{0.7}{1.05}= 0.67[/tex]
The probability that a random sample of 16 hobbits have a mean height of no more than 36.7 in tall is P=0.74857.
[tex]P(x_{16}\leq36.7)=P(z\leq 0.67)=0.74857[/tex]
c) We apply the same principle as in pont b.
We can calculate z with the sample standard deviation
[tex]z=\frac{x-\mu}{\sigma/\sqrt{n}}=\frac{36.7-36}{4.2/\sqrt{100}}=\frac{0.7}{0.42}= 1.67[/tex]
The probability that a random sample of 100 hobbits have a mean height of no more than 36.7 in tall is P=0.95254.
[tex]P(x_{100}\leq36.7)=P(z\leq 1.67)=0.95254[/tex]
