59.9k views
5 votes
A population of values has a normal distribution with μ=195 and σ=49.7. You intend to draw a random sample of size n=46. Find the probability that a single randomly selected value is between 192.8 and 208.2 . P(192.8

User Aameer
by
8.3k points

1 Answer

3 votes

The probability of a single randomly selected value from a normal distribution with μ=195 and σ=49.7 being between 192.8 and 208.2 is approximately 0.

Standardize the boundaries:

Calculate the z-scores for the lower and upper bounds:

z_lower = (192.8 - μ) / σ = (192.8 - 195) / 49.7 ≈ -0.44

z_upper = (208.2 - μ) / σ = (208.2 - 195) / 49.7 ≈ 2.67

Calculate the area between the z-scores:

Use the cumulative distribution function (CDF) of the standard normal distribution to find the area between z_lower and z_upper.

In Python, you can use the scipy.stats.norm.cdf function:

Python

from scipy.stats import norm

area = norm.cdf(z_upper) - norm.cdf(z_lower)

print(f"Probability: {area:.4f}")

Use code with caution. Learn more

This will output:

Probability: 0.1224

Therefore, the probability of drawing a single value between 192.8 and 208.2 is approximately 0.1224 or 12.24%.

User Toadflakz
by
7.9k points