ECON1280-CH6

Distribution of sample means

*As the sample size increases, the distribution of sample means tends to approach a normal distribution. *As the sample size increases, the dotplots become narrower, showing that the standard deviation of the sample means becomes smaller.

Manual Construction of a Normal Quantile Plot

1. Lowest to highest 2. 𝑥(𝑖) =𝑖 - 0.5/𝑛 3. find the z scores corresponding to the cumulative left areas as y 4. match original data and corresponding z-score y 5. whether straight?

Assessing normality

1. histogram-bell-shape 2. 1- outliers 3. A normal quantile plot: x value is from the original set of sample data, and y value is the corresponding z score that is a quantile value expected from the standard normal distribution. Straight line. * Reject if the points do not lie reasonably close to a straight line or the points show some systematic pattern that is not a straight-line pattern.

Data transformation

1. log(x+1): lognormal distribution 2. 1/x, x^1/2, x^2 * for being normal or correct other deficiencies (such as same variance in different samples required)

Normal distribution

A continuous random variable has a distribution with a graph that is symmetric and bell-shaped which can be described by the equation.

Uniform distribution

A continuous random variable's values are spread evenly over the range of possibilities.

Unbiased estimator

A statistic that targets the value of the population parameter.

Estimator

A statistic used to infer the value of a population parameter.

Variance

E(x^2)-E(X)^2

Critical Values

For a normal distribution, it is a z score on the borderline separating the z scores that are likely to occur from those that are unlikely. *Common critical values are z = -1.96 and z = 1.96, 2.5% and 97.5%

Central Limit Theorem

For a population with any distribution, the distribution of the sample means approaches a normal distribution as the sample size increases. (to estimate population parameter) *n>30 or original normal distribution *Mean of sample means: miu *Standard deviation of sample means: sigma/n^1/2=standard error of the mean.

Standard normal distribution

It is a normal probability distribution with parameters miu=0 and sigma=1

Sampling distribution of a statistic

It is the distribution of all values of the statistic when all possible samples of the same size n are taken from the same population. *Normal distribution: mean proportion *Skewed-right: variance *Unbiased estimators: mean, proportion, variance *Biased estimators: median, range, standard deviation (bias is small when sample are large)

Density curve

It is the graph of a continuous probability distribution. * Total area under the curve = 1 * Every point on the curve can't have a negative height.

Expected value

It is weighted average of the possible value of x, weighted by probability. if 𝑋 is discrete if 𝑋 is continuous: integrity

Description

The sampling distribution of the sample mean is a normal distribution with miu = 100 and sigma = 15.

finite population correction factor

When sampling without replacement and the sample size n is greater than 5% of the finite population size N (that is, n>0.05N ), adjust the standard deviation of sample means by multiplying it by the finite population correction factor:

Notation

z alpha: The expression za denotes the z score with an area of a to its right. p = population proportion p(hat) = sample proportion miu x-bar=the mean of the sample means sigma x-bar= the standard deviation of the sample means

Converting to a Standard Normal Distribution

z=x-miu/sigma x=miu+(z*sigma)