Question

The overhead reach distances of adult females are normally distributed with a mean of

202.5 cm202.5 cm

and a standard deviation of

8 cm8 cm.

a. Find the probability that an individual distance is greater than

215.00215.00

cm.b. Find the probability that the mean for

2020

randomly selected distances is greater than 201.00 cm.201.00 cm.

c. Why can the normal distribution be used in part​ (b), even though the sample size does not exceed​ 30?

a. The probability is

nothing.

​(Round to four decimal places as​ needed.)

b. The probability is

nothing.

​(Round to four decimal places as​ needed.)

c. Choose the correct answer below.

A.

The normal distribution can be used because the finite population correction factor is small.

B.

The normal distribution can be used because the mean is large.

C.

The normal distribution can be used because the original population has a normal distribution.

D.

The normal distribution can be used because the probability is less than 0.5

Click to select your answer(s).

Answer 1

Let X: The overhead reach distances of adult females

Given, X follows normal distribution with mean = 202.5 cm and standard deviation = 8 cm , i.e.,

a.) To find the probability that an individual distance is greater than 215 cm i.e.,

P(X>215)

Using Central Limit Theorem, which states that,

Thus,

  

  

  

   ..........................(using standard normal table)

  

Therefore, probability that an individual distance is greater than 215 cm is 0.0594

b.) To find the probability that the mean for 20 randomly selected distances is greater than 201.00 cm, i.e., P( > 201)

Here, given n=20 and   = 201

We know that,

Now, using Central Limit Theorem for Mean , which states that,

Thus,

  

  

  

  

  

  

Hence, the probability that the mean for 20 randomly selected distances is greater than 201.00 cm is 0.7969

c.) The option (C) i.e.,The normal distribution can be used because the original population has a normal distribution is the correct option since, Central Limit Theorem is applicable even for sample size less than 30 only when the population is normally distributed.