Lecture 5 The Normal Distribution
Lecture 5 The Normal Distribution 1. The Normal Distribution A Continuous Probability Distribution Function (pdf)
-‐ -‐ -‐ -‐
Bell shaped and symmetrical about the mean Location is determined by the mean, µ Spread is determined by the standard deviation, 𝜎 The random variable X has an infinite theoretical range from -‐∞ to +∞
2. The Normal Distribution Shape Changing µ shifts the distribution left or right
Changing 𝜎 increases or decreases the spread
3. Standardized Normal Distribution (Z) 𝒁=
𝑿−𝝁 𝝈
4. Finding Normal Probabilities Probability = Area Under the Curve The total area under the curve is 1 and the curve is symmetrical, so half the area is either side of the mean.
5. General Procedure for Finding Probabilities for a Normally Distributed Variable, X 1) Draw the normal curve for the problem in terms of X 2) Transform the X-‐values into Z-‐values 3) Use the Z table “tool” to find the required area
Lecture 6 Sampling distribution of the sample mean & sampling distribution of the sample proportion
1. A Sampling Distribution The distribution of all possible values of a statistic, using the same sample size, selected from a population 2. Standard Error of the Mean A measure of the variability of the mean from sample to sample is given by what is called the Standard Error of the Mean, 𝜎) 𝝈 𝝈𝑿 = 𝒏 -‐ The standard error of the mean decreases as the sample size increases -‐ A key point: If a population variable is normally distributed with mean µ and standard deviation σ, the sampling distribution of 𝑋 is also exactly normally distributed with : 𝝈 𝝁𝑿 = 𝝁 𝒂𝒏𝒅 𝝈𝑿 = 𝒏 3. Z-‐value for Sampling Distribution of the Sample Mean 𝑿−𝝁 𝑿−𝝁 𝒁= = 𝝈 𝝈𝑿 𝒏 4. Sampling Distribution properties -‐ Sampling distribution of the sample mean if exactly normally distributed
-‐ As n increases, σ0 decreases
-‐ -‐ -‐ -‐
Bell shaped and symmetrical about the mean Location is determined by the mean, µ Spread is determined by the standard deviation, 𝜎 The random variable X has an infinite theoretical range from -‐∞ to +∞
2. The Normal Distribution Shape Changing µ shifts the distribution left or right
Changing 𝜎 increases or decreases the spread
3. Standardized Normal Distribution (Z) 𝒁=
𝑿−𝝁 𝝈
4. Finding Normal Probabilities Probability = Area Under the Curve The total area under the curve is 1 and the curve is symmetrical, so half the area is either side of the mean.
5. General Procedure for Finding Probabilities for a Normally Distributed Variable, X 1) Draw the normal curve for the problem in terms of X 2) Transform the X-‐values into Z-‐values 3) Use the Z table “tool” to find the required area
Lecture 6 Sampling distribution of the sample mean & sampling distribution of the sample proportion
1. A Sampling Distribution The distribution of all possible values of a statistic, using the same sample size, selected from a population 2. Standard Error of the Mean A measure of the variability of the mean from sample to sample is given by what is called the Standard Error of the Mean, 𝜎) 𝝈 𝝈𝑿 = 𝒏 -‐ The standard error of the mean decreases as the sample size increases -‐ A key point: If a population variable is normally distributed with mean µ and standard deviation σ, the sampling distribution of 𝑋 is also exactly normally distributed with : 𝝈 𝝁𝑿 = 𝝁 𝒂𝒏𝒅 𝝈𝑿 = 𝒏 3. Z-‐value for Sampling Distribution of the Sample Mean 𝑿−𝝁 𝑿−𝝁 𝒁= = 𝝈 𝝈𝑿 𝒏 4. Sampling Distribution properties -‐ Sampling distribution of the sample mean if exactly normally distributed
-‐ As n increases, σ0 decreases
