Chapter 7 Statistical Inference

Statistical Inference

focuses on drawing conclusions about populations from samples. Statistical inference includes estimation of population parameters and hypothesis testing

hypothesis testing

which is a technique that allows you to draw valid statistical conclusions about the value of population parameters or differences among them.

Null Hypothesis (H0)

describes the existing theory or a belief that is accepted as valid unless strong statistical evidence exists to the contrary. can never be proven true only false or fail to reject

alternative hypothesis

is the complement of the null hypothesis; it must be true if the null hypothesis is false

The null hypothesis is denoted by H0, and the alternative hypothesis is denoted by H1. Using sample data, we either

1.) reject the null hypothesis and conclude that the sample data provide sufficient statistical evidence to support the alternative hypothesis, or

2.) fail to reject the null hypothesis and conclude that the sample data do not support the alternative hypothesis.

Hypothesis-Testing Procedure

Conducting a hypothesis test involves several steps:

1.) Identifying the population parameter of interest and formulating the hypotheses to test

2.) Selecting a level of significance, which defines the risk of drawing an incorrect conclusion when the assumed hypothesis is actually true

3.) Determining a decision rule on which to base a conclusion

4.) Collecting data and calculating a test statistic

5.) Applying the decision rule to the test statistic and drawing a conclusion

One sample hypothesis tests

H0: population parameter _> constant vs H1: population parameter < constant

H0: population parameter _< vs. H1: population parameter > constant

H0: population parameter = constant vs. H1: population parameter not equal to constant

Hypothesis testing always assumes the null hypothesis

is true. We use sample data to determine whether the alternative hypothesis is likely to be true

If we FTR the null hypothesis, we

we do not have sufficient evidence to reject it

Hypothesis testing 4 outcomes

1.) The null hypothesis is actually true, and the test correctly fails to reject it.

2.) The null hypothesis is actually false, and the hypothesis test correctly reaches this conclusion.

3.) The null hypothesis is actually true, but the hypothesis test incorrectly rejects it (called Type I error).

4.) The null hypothesis is actually false, but the hypothesis test incorrectly fails to reject it (called Type II error).

Level of significance

the probability of making a type I error denoted by alpha

Commonly used levels for alpha

• 0.10 (90%)

• 0.05 (95%)

• 0.01 (99%)

confidence coefficient

The confidence level expressed as a decimal value. For example, .95 is the confidence coefficient for a 95% confidence level.

Type II error

failing to reject a false null hypothesis. cannot control probability

power of the test

Probability of correctly rejecting a false null hypothesis.

test statistic formula

Z= (x bar - mu)/sigma/SQRT of N

t= x bar - mu/sample/SQRT of N

You use a Z test if sigma is known, if sigma is unknown then t test

The decision to reject or fail to reject a null hypothesis is based on computing

a test statistic

The conclusion to reject or fail to reject H0 is based on comparing the value of the test statistic to a

" Critical value" found in excel

two-tailed test of hypothesis

The rejection region occurs in both the upper and lower tail of the distribution

One tail tests

a test with the rejection region in only one tail of the distribution

< = lower

= upper (1 -norm.dist)

Summary of One-Sample Hypothesis Tests for the Mean

Case 1: (σ Unknown)

Determine whether the proper hypotheses represent a lower-tailed, upper-tailed, or two-tailed test.

Calculate the test statistic using formula (7.2).

Find the critical value.

If it is a lower-tailed test, the critical value is found using the Excel function T.INV(1−α, n−1). Note the minus sign!

If it is an upper-tailed test, the critical value is found using the Excel function T.INV(1−α, n−1).

If you have a two-tailed test, use T.INV.2T(α, n−1); the critical values will be both positive and negative.

Compare the test statistic to the critical value(s) and draw a conclusion to either reject the null hypothesis or fail to reject it.

Summary of One-Sample Hypothesis Tests for the Mean case 2

Case 2: (σ Known)

Determine whether the proper hypotheses represent a lower-tailed, upper-tailed, or two-tailed test.

Calculate the test statistic using formula (7.1).

Find the critical value. The critical value for a one-sample, one-tailed test when the standard deviation is known is the value of the normal distribution that has a tail area of alpha. This may be found by using Table A.1 in Appendix A to find the z-value

corresponding to an area of 1−α or using the Excel function NORM.S.INV(1−α). Remember that for a lower-tailed test, the critical value is negative.

For a two-tailed test, use α/2.

If it is a lower-tailed test, the critical value is found using the Excel function NORM.S.INV(1−α). Note the minus sign!

If it is an upper-tailed test, the critical value is found using the Excel function NORM.S.INV(1−α).

If you have a two-tailed test, use NORM.S.INV(1−α/2); the critical values will be both positive and negative.

Compare the test statistic to the critical value(s) and draw a conclusion to either reject the null hypothesis or fail to reject it.

p-value (observed significance level)

An alternative approach to comparing a test statistic to a critical value in hypothesis testing is to find the probability of obtaining a test statistic value equal to or more extreme than that obtained from the sample data when the null hypothesis is true

if whenever p<α, reject the null hypothesis and otherwise fail to reject it

One Sample Proportion Test

z= (p hat - pie null)/SQRT(pie null(1- pie null)/n

Two-Sample Hypothesis Tests

In a two-sample test for differences in means, we always test hypotheses of the form

(7.4)

H0: μ1−μ2{≥,≤, or=} 0

H1: μ1−μ2{

The null hypothesis for ANOVA is that the population means of all m groups are equal; the alternative hypothesis is that at least one mean differs from the rest:

H0: μ1=μ2=⋯=μmH1: at least one mean is different from the others

ANOVA 3 Assumptions

1.) are randomly and independently obtained,

2.) are normally distributed, and

3.) have equal variances.

Analysis of variance (ANOVA)

a statistical technique that determines whether three or more means are statistically different from one another

Factor

the variable of interest