Question

The upper leg length of 20 to 29-year-old males is normally distributed with a mean length of 43.7 cm and a standard deviation of 4.2 cm. a random sample of 9 males who are 20 to 29 years old is obtained. what it the probability that the mean leg length is less than 20 cm? 0.1894 pratically 0 0.2134 0.7898

Answer 1

The probability that the mean leg length is less than 20 cm is practically 0.

Step-by-step explanation:

To find the probability that the mean leg length is less than 20 cm, we can use the sampling distribution of the sample mean. The sampling distribution of the sample mean is approximately normal when the sample size is large enough. In this case, the sample size is 9, which is smaller than 30 but still reasonably large, so we can assume that the sampling distribution of the sample mean follows a normal distribution.

We can standardize the sample mean using the formula:

z = (x - μ) / (σ / sqrt(n))

Where:

• z: the z-score
• x: the value of the sample mean
• μ: the population mean
• σ: the population standard deviation
• n: the sample size

Substituting the given values:

z = (20 - 43.7) / (4.2 / sqrt(9))

z = -23.7 / (4.2 / 3)

z = -23.7 / 1.4

z ≈ -16.93

Looking up the z-score in a standard normal distribution table, we find that the probability of getting a z-score less than -16.93 is practically 0. Therefore, the probability that the mean leg length is less than 20 cm is practically 0.

Answer 2

Given:
μ = 43.7 cm, the population mean
σ = 4.2 cm, the population standard deviation.

We want to test against the population statistics with
n = 9, the sample size,
x = 20 cm, the random variable.
We want to find P(x < 20).

Calculate the z-score.
z = (x - μ)/σ
= (20 - 43.7)/4.2
= -5.643

From the standard tables, obtain
P(z < -5.643) = 0 (actually about 8.5 x 10⁻⁹)

Answer: Practically zero.