Question

The overhead reach distances of adult females are normally distributed with a mean of 202.5 cm and a standard deviation of 8.9 cm.

a. Find the probability that an individual distance is greater than 212.50 cm
b. Find the probability that the mean for 25 randomly selected distances is greater than 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
(Round to four decimal places as needed)
b. The probability is I
(Round to four decimal places as needed.)
c. Choose the correct answer below.
O A The normal distribution can be used because the mean is large.
B. The normal distribution can be used because the finite population correction factor is small.
OC. The normal distribution can be used because the probability is less than 0.5
OD. The normal distribution can be used because the original population has a normal distribution

Answer 1

I am not sure if you're accustomed to using a z-score and table of probabilities or technology, so I will answer using both!

a) Since you're choosing one person from the population, you will use the mean and standard deviation from the population. z= (210.9-197.5)/8.6 = 1.56 Using a table of normal probabilities,

P(x>210.9)= 1-P(z<1.56)= 1- 0.9406=0.0594

Using technology: normalcdf(210.9, 1E99, 197.5, 8.6)=0.0596

b) Since you're calculating the population for a sample to occur, you need to get a mean of the sampling distribution and standard deviation of the sampling distribution.

Meanpop=Meansampling dist.=197.5

St. Devsampling dist.= Stdpopulation/√n = 8.6/√15 = 2.22

Using technology, P(X>196.20)= normalcdf( 196.20, 1E99, 197.5, 2.22) = 0.7209

c) You only have to worry about the sample size in a sampling distribution if you either do not know the shape of the distribution of the population or if the distribution of the population is not normally distributed. Since this population is normally distributed (as stated in the given information), then your sample size does not have to be greater than 30.

Explanation: