Question

the overhead reach distances of adult females are normally distributed with a mean of 205.5 cm and a standard deviation of 7.8 cm.
a. find the probability that an individual distance is greater than 218.90 cm.
(round to four decimal places as needed.)
the probability is .0427
b. find the probability that the mean for 25 randomly selected distances is greater than 204.00 cm.
(round to four decimal places as needed.)
the probability is .8315
c. choose the correct answer below.
o a. the normal distribution can be used because the finite population correction factor is small.
o b. the normal distribution can be used because the original population has a normal distribution.
o c. the normal distribution can be used because the probability is less than 0.5
o d. the normal distribution can be used because the mean is large.

Explanation:

Step1: Calculate z - score for part (a)

The formula for the z - score is \(z=\frac{x-\mu}{\sigma}\). Given \(\mu = 205.5\), \(\sigma=7.8\), and \(x = 218.9\).

$$z=\frac{218.9 - 205.5}{7.8}=\frac{13.4}{7.8}\approx1.72$$

Using the standard normal table \(P(X>218.9)=1 - P(Z\leq1.72)\). From the standard normal table \(P(Z\leq1.72)=0.9573\), so \(P(X > 218.9)=1-0.9573 = 0.0427\)

Step2: Calculate z - score for part (b)

The formula for the z - score of the sample mean is \(z=\frac{\bar{x}-\mu}{\frac{\sigma}{\sqrt{n}}}\). Given \(\mu = 205.5\), \(\sigma = 7.8\), \(n = 25\), and \(\bar{x}=204\)

$$z=\frac{204 - 205.5}{\frac{7.8}{\sqrt{25}}}=\frac{- 1.5}{\frac{7.8}{5}}=\frac{-1.5}{1.56}\approx - 0.96$$

Using the standard normal table \(P(\bar{X}>204)=1 - P(Z\leq - 0.96)\). Since \(P(Z\leq - 0.96)=0.1685\), then \(P(\bar{X}>204)=1 - 0.1685=0.8315\)

Step3: Justify the use of normal distribution in part (b)

The Central Limit Theorem states that if the population is normally distributed, the sampling distribution of the sample mean \(\bar{X}\) is also normally distributed for any sample size \(n\). Here, the original population of adult female overhead reach distances is normally distributed.

Answer:

a. \(0.0427\)
b. \(0.8315\)
c. B. The normal distribution can be used because the original population has a normal distribution.