Chapter 6 Normal Distributions & Sampling

Density Curve

Models the probability distribution of a continuous random variable with a curve that:

1) is always on or above the x-axis, and

2) has an area of exactly 1 underneath

Equal-Areas Point

The median of a density curve; it is the point that divides the area under the curve in half

Normal Distribution

Described by a symmetric, single-peaked, bell-shaped density curve called a Normal Curve

What 2 characteristics completely specify a Normal Dsitribution?

1. The mean “μ”

2. The standard deviation “σ”

T/F: The mean “μ” of a Normal Distribution is located at the midpoint and is the same as the median

True

The Empirical Rule

States that in any normal distribution about 68% of values fall within 1σ, about 95% fall within 2σ, and about 99.7% fall within 3σ of the mean.

Standard Normal Distribution

The normal distribution with mean 0 and standard deviation 1, which can be used to find areas under any normal curve

Statistic

A number that describes some characteristic of a sample

Parameter

A number that describes some characteristic of the population

The value of a _____ is usually unknown. To estimate its value, we use _____ calculated from a random sample.

Parameter; Statistic

Sampling Distribution

The distribution of values taken by the statistic in all possible samples of the same size from the same population

Sampling Distribution of the Sample Proportion (p̂)

Describes the distribution of values taken by the sample proportion (p̂) in all possible samples of the same size from the same population

To determine is the Sample Proportion (p̂) provides convincing evidence, we need to know the _________________.

Sampling distribution of the sample proportion (p̂)

How to calculate the Mean of the Sample Proportion (μ_p)

μ_p = p

How to calculate the Standard Deviation of the Sampling Distribution (σ_p)

σ_p = sqrt((p(1-p)/n)

What are 4 identifying facts about the Sample Proportion (p̂)

1. The sample proportion (p̂) is an unbiased estimator of the population proportion (the mean “μ_p” is equal to the population proportion “p”)

2. The standard deviation of the sampling distribution of p̂ describes the typical distance between p̂ and the population proportion “p”

3. The sampling distribution of p̂ is less variable for larger samples

4. The formula for the standard deviation σ_p requires that the observations are independent

Large Counts Condition

Says that the sampling distribution of p̂ will be approximately normal when np >= 10 and n(1-p) >= 10

Sampling Distribution of the Sample Mean (x̂)

Describes the distribution of values taken by the sample mean x̂ in all possible samples of the same size from the same population

When sample data are quantitative, we use the ____ ____ to estimate the mean “μ” of a population

Sample Mean (x̂)

How to calculate the Mean of the Sampling Distribution (μ_x)

μ_x = μ

How to calculate the Standard Deviation for the Sampling Distribution (σ_x)

σ_x = σ/sqrt(n)