Stat: Chapter 6

Procedure for Finding Areas with a Nonstandard Normal Distribution

1. Sketch a normal curve, label the mean and any specific x values, then shade the region representing the desired probability. 2. For each relevant value x that is a boundary for the shaded region, use Formula 6-2 to convert that value to the equivalent z score. 3. Use computer software or a cal or table A-2 to find the area of the shaded region. This area is the desired probability.

Procedure for Determining Whether It Is Reasonable to Assume That Sample Data Are from a Population Having a Normal Distribution

1 Histogram: construct a histogram. If the histogram departs dramatically from a bell shape, conclude that the data do not have a normal distribution. 2. Outliers: Identify outliers. If there is more than one outlier present, conclude that the data do not have a distribution. (Just one outlier could be an error or the result of chance variation, but be careful, because even single outlier can have a dramatic effect on results.) 3. Normal quantile plot: If the histogram is basically symmetric and the number of outliers is 0 or 1, use technology to generate a normal quantile plot. Apply the following criteria to determine whether or not the distribution is normal. (These criteria can be used loosely for small samples, but they should be used more strictly for large samples.)

Considerations for Practical Problem solving

1. Checking requirements: When wor

Procedure for Finding a z Score from a Known Area

1. Draw a bell-shaped curve and identify the region under the curve that corresponds to the given probability. If that region is not a cumulative region from the left, work instead with a known region that is a cumulative region from the left. 2. Use the cumulative area from the left, locate the closest probability in the body of the table, and identify the corresponding z score.

Behavior of Sample Proportion

1. Sample proportions target the value of the population proportion. (That is, the mean of the sample proportions is the population proportion. The expected value of the sample proportion is equal to the population proportion.) 2. The distribution of sample proportions tends to approximate a normal distribution.

Uniform distribution key properties

1. The area under the graph of a probability distribution is equal to 1. 2. There is a correspondence between area and probability (or relative frequency), so some probabilities can be found by identifying the corresponding areas in the graph.

uniform distribution

A continuous random variable has a uniform distribution if its values are spread evenly over the range of possibilities. The graph of a uniform distribution results in a rectangular shape. Also called a density curve. P(x)- vertical axis, x- horizontal axis

Standard Normal Distribution

A normal distribution with the parameters of μ = 0 and σ =1. The total area under its density curve is equal to 1.

Sampling Distribution of a Statistic

Such as sample mean or sample proportion) is the distribution of all values of the statistic when all possible samples if the same size n are taken from the same population. (The sampling distribution of a statistic is typically represented as a probability distribution in the format of a table, probability histogram, or formula.)

Estimator

is a statistic used to infer (estimate) the value of a population parameter.

Normal Distribution formaula

y = [(e^-1/2(x-μ/σ)^2)/(σ√2π)]

Distribution of sample means

μ sub x̄. It is essentially an average of averages.

Normal Quantile Plot

(Or normal probability plot) is a graph of points (x,y) where each x value is from the original set of sample data, and each y value is the corresponding z score that is a quantile value expected from the standard normal distribution.

Behavior of Sample Means

1. The sample means target the value of the population mean. (That is, the mean of the sample means is the population mean. The expected value of the sample mean is equal to the population mean.) 2. The distribution of sample means tends to be a normal distribution. (This will be discussed further in the following section, but the distribution tends to become closer to a normal distribution as the sample size increases.)

Behavior of Sample Variances

1. The sample variables target the value of the population variance. (That is, the mean of the sample variances is the population variance. The expected value of the sample variance is equal to the population variance.) 2. The distribution of sample variances tends to be a distribution skewed to the right.

Central Limit Theorem

For all samples of the same size n with n > 30, the sampling distribution of x̄ can be approximated by a normal distribution with mean μ and standard deviation σ/√n.

Normal Distribution

If a continuous random variable has a distribution with a graph that is symmetric and bell-shaped,as the image, and it can be described by the equation given as by the formula

RareEvent Rule for Inferential Statistics

If, under a given assumption, the probability of a particular observed even is extremely small (such as less than 0.05), we conclude that the assumption is probably not correct.

Unbiased Estimator

Is a statistic that targets the value of the population parameter in the sense that the sampling distribution of the statistic has a mean that is equal to the mean of the corresponding parameter.

Sampling Distribution of the Sample Proportion

Is the distribution of sample proportions, with all samples having the same sample size n taken from the same population. Notation: p = population proportion p̂ = sample proportion

Density Curve

Must satisfy two requirements: 1. The total area under the curve must equal 1. 2. Every point on the curve must have a vertical height that is 0 or greater. (That is, the curve cannot fall below the x-axis.) *Because the total area under the density curve is equal to 1, there is a correspondence between area and probability. P(x)-vertical axis, x-horizontal axis.

Notation for the Sampling Distribution of x̄

Of all possible simple random samples of size n are selected from a population with mean μ and standard deviation σ , the mean of the sample means is denoted by μ-sub-x̄ and the standard deviation of all samples means is denoted by σ-sub-x̄. (σ-sub-x̄ is called the standard error of the mean.) Mean of all values of x̄: μ-sub-x̄ = μ Standard deviation of all values of x̄: σ-sub-x̄ = σ/√x

Normal Distribution for Pop.

The population distribution is normal if the pattern of the points is reasonably close to a straight line ad the points do not show some systematic pattern that is not a straight-line pattern.

Not a Normal Distribution

The population distribution is not normal if either or both of these two conditions applies: Points do not lie reasonably close to a straight line. The points show some systematic pattern that is not a straight line pattern.

Unbiased Estimators stats

These statistics are unbiased estimators. That is, they each target the value of the corresponding population parameter: Mean x̄ Variance s² Proportion p̂

Biased Estimators Stats

These statistics are unbiased estimators. That is, they each target the value of the corresponding population parameter: Median Range Standard Deviation s* *The sample standard deviations do not target the population standard deviation σ, but the biased is relatively small in large samples, so s is often used to estimate σ even though s is a biased estimator of σ.

Continuity Correction

When we use the normal distribution (which is a continuous probability distribution) as an approximation to the binomial distribution (which is discrete), a continuity correction is made to a discrete whole number x in the binomial distribution by representing the discrete whole number x by the interval from x-0.5 to x+0.5

Application of Z Score

When working with a normal distribution that is nonstandard (with a mean different from 0 and/or a standard deviation different from 1), we use formula 6-2 to transform a value x to a z score, then we proceed with the same methods from sec.6.2 z = (x - μ)/σ

Critical Value

a z score separating unlikely values from those that are likely to occur.

Sampling distribution of the sample mean

is the distribution of all possible sample means (or the distribution of the variable x̄), with all samples having the same sample size n taken from the sample population. (The sampling distribution of the sample mean is typically represented as a probability distribution in the format of a table, probability histogram, or formula.)

Sampling Distribution of the Sample Variance

is the distribution of sample variances, with all samples having the same sample size n taken from the same population. (The sampling distribution of the sample variance is typically represented as a probability distribution in the format of a table, probability histogram, or formula.)