Question

the overhead reach distances of adult females are normally distributed with a mean of 205.5 cm and a standard deviation of 8 cm.
a. find the probability that an individual distance is greater than 214.80 cm.
b. find the probability that the mean for 25 randomly selected distances is greater than 204.20 cm.
c. why can the normal distribution be used in part (b), even though the sample size does not exceed 30?
(round to four decimal places as needed.)
b. the probability is 0.7917
(round to four decimal places as needed.)
c. choose the correct answer below.
○ a. the normal distribution can be used because the probability is less than 0.5
○ b. the normal distribution can be used because the original population has a normal distribution.
○ c. the normal distribution can be used because the finite population correction factor is small.
○ d. the normal distribution can be used because the mean is large

Explanation:

Step1: Calculate the z - score for part (a)

The formula for the z - score is \(z=\frac{x-\mu}{\sigma}\), where \(x = 214.80\), \(\mu=205.5\), and \(\sigma = 8\).

$$z=\frac{214.80 - 205.5}{8}=\frac{9.3}{8}=1.1625$$

We want \(P(X>214.80)\), which is \(1 - P(X\leq214.80)\). Using the standard normal table, \(P(Z\leq1.1625)\approx0.8770\). So \(P(X>214.80)=1 - 0.8770 = 0.1230\)

Step2: Calculate the z - score for part (b)

The formula for the z - score of the sample mean is \(z=\frac{\bar{x}-\mu}{\frac{\sigma}{\sqrt{n}}}\), where \(\bar{x}=204.20\), \(\mu = 205.5\), \(\sigma=8\), and \(n = 25\)

$$z=\frac{204.20-205.5}{\frac{8}{\sqrt{25}}}=\frac{- 1.3}{\frac{8}{5}}=\frac{-1.3}{1.6}=-0.8125$$

We want \(P(\bar{X}>204.20)\), which is \(1 - P(\bar{X}\leq204.20)\). Using the standard normal table, \(P(Z\leq - 0.8125)\approx0.2083\). So \(P(\bar{X}>204.20)=1 - 0.2083=0.7917\)

Step3: Answer part (c)

The Central Limit Theorem states that if the population is normally distributed, then the sampling distribution of the sample mean \(\bar{X}\) is also normally distributed for any sample size \(n\). Since the original population of overhead reach distances of adult females is normally distributed, the normal distribution can be used for the sampling distribution of the sample mean in part (b) even though \(n = 25<30\)

Answer:

a. \(0.1230\)
b. \(0.7917\)
c. B. The normal distribution can be used because the original population has a normal distribution.