Question

Asked Nov 18, 2019

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The lengths of adult males' hands are normally distributed with mean 192 mm and standard deviation is 7.4 mm. Suppose that 40 individuals are randomly chosen. Round all answers to 4 where possible.

- What is the distribution of ¯xx¯? ¯xx¯ ~ N(,)
- For the group of 40, find the probability that the average hand length is more than 193.
- Find the first quartile for the average adult male hand length for this sample size. Round to 2 decimal places.
- For part b), is the assumption that the distribution is normal necessary? NoYes

Step 1

**Central Limit Theorem for mean:**

If a random sample of size *n *is taken from a population having mean µ and standard deviation σ then, as the sample size increases, the sample mean approaches the normal distribution with mean µ and standard deviation σ/sqrt(n).

Step 2

**1.**

**Finding the distribution of X-bar:**

It is given that the lengths of adults’ male hands follows normal distribution with mean 192mm and standard deviation 7.4mm.

That is, µ= 192, σ = 7.4.

Random samples of 40 (*n*) individuals is considered.

Denote the random variable *X* as the length of hand for a randomly selected individual.

By central limit theorem for mean, the average hand length follows a normal distribution with mean μ = 192 and standard deviation 7.4/sqrt (40) = 1.1700.

Thus, the distribution of **X-bar ~ N(192,1.17).**

Step 3

**2.**

**Finding the probability that the average hand length is more than 193:**

The probability that the aver...

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