Question

From statements in J. R. R. Tolkien's "The Fellowship of the Ring" (Prologue I, paragraph 4), I construe that the heights of hobbits are approximately normally distributed with mean 36 inches and standard deviation 4.2 inches. Find the following probabilities (be as accurate as our process allows).
A. Find the probability that one hobbit picked at random is no more than 36.7 inches tall.
B. Find the probability that a simple random sample of 16 hobbits have a mean height of no more than 36.7 inches tall.
C. Find the probability that a simple random sample of 100 hobbits have a mean height of no more than 36.7 inches tall.
D. Find the probability that a simple random sample of 400 hobbits have a mean height of no more than 36.7 inches tall.

Answer 1

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.

$z=\frac{x-\mu}{\sigma}=\frac{36.7-36}{4.2}=\frac{0.7}{4.2}=0.167$

The probability that a hobbit picked at random is no more than 36.7 in tall is P= 0.56631.

$P(x\leq36.7)=P(z\leq 0.167)=0.56631$

b) In this case, the sample standard deviation change to

$\sigma_s=\frac{\sigma}{\sqrt{n} }$.

We can calculate z with the sample standard deviation

$z=\frac{x-\mu}{\sigma/\sqrt{n}}=\frac{36.7-36}{4.2/\sqrt{16}}=\frac{0.7}{1.05}= 0.67$

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.

$P(x_{16}\leq36.7)=P(z\leq 0.67)=0.74857$

c) We apply the same principle as in pont b.

We can calculate z with the sample standard deviation

$z=\frac{x-\mu}{\sigma/\sqrt{n}}=\frac{36.7-36}{4.2/\sqrt{100}}=\frac{0.7}{0.42}= 1.67$

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.

$P(x_{100}\leq36.7)=P(z\leq 1.67)=0.95254$