Normal Distribution

The normal distribution is a continuous probability distribution with a bell-shaped curve. It is characterized by its mean (μ) and standard deviation (σ). The standard normal distribution has μ=0 and σ=1, with Z = (X - μ)/σ. The empirical rule (68-95-99.7 rule) states that 68% of data falls within ±1σ, 95% within ±2σ, and 99.7% within ±3σ of the mean. The normal distribution is widely used due to the Central Limit Theorem.

Governing FormulaZ = (X - μ)/σ, f(x) = (1/(σ√2π)) × e^(-(x-μ)²/(2σ²))

Knowledge Check

10 Questions

1.Scores on a test are normally distributed with mean 70 and standard deviation 10. What is the Z-score for a student who scored 85?

2.In a standard normal distribution, what is the probability that Z is less than 1.96?

3.According to the empirical rule, what percentage of data falls within ±2 standard deviations of the mean?

4.A population has mean 100 and standard deviation 15. What is the probability that X > 130? (Z = 2, P(Z>2) = 0.0228)

5.What is the mean of the standard normal distribution?

6.The Central Limit Theorem states that:

7.If Z = -1.5, what is the probability that Z is greater than -1.5?

8.A product's weight is normally distributed with mean 50 g and standard deviation 2 g. What weight corresponds to the 84th percentile? (Z = 1.0 for 84th percentile)

9.What is the standard deviation of the standard normal distribution?

10.In a normal distribution, the mean, median, and mode are: