Statistical Inference

Type I error α (significance level)

A Type I error occurs when the researcher rejects a null hypothesis when it is true. The probability of committing a Type I error is called the significance level. This probability is also called alpha, and is often denoted by α.

Type II error β (Power of the test)

A Type II error occurs when the researcher fails to reject a null hypothesis that is false. The probability of committing a Type II error is called Beta, and is often denoted by β. The probability of not committing a Type II error is called the Power of the test.

Interval estimate = a < x < b

An interval estimate is defined by two numbers, between which a population parameter is said to lie. For example, a < x < b is an interval estimate of the population mean μ. It indicates that the population mean is greater than a but less than b.

Hypothesis testing Step 4: Interpret results.

Apply the decision rule described in the analysis plan. If the value of the test statistic is unlikely, based on the null hypothesis, reject the null hypothesis.

What is Margin of Error?

In a confidence interval, the range of values above and below the sample statistic

A confidence interval for µ is 13 ± 5. The value of 5 in this expression is the estimate's standard error/

False it is the margin of error

Hypothesis testing Step 3: Analyze sample data.

Find the value of the test statistic (mean score, proportion, t statistic, z-score, etc.) described in the analysis plan.

Example of how to interpret a confidence interval

For example, suppose we compute an interval estimate of a population parameter. We might describe this interval estimate as a 95% confidence interval. This means that if we used the same sampling method to select different samples and compute different interval estimates, the true population parameter would fall within a range defined by the sample statistic + margin of error 95% of the time.

In statistics, what is estimation?

In statistics, estimation refers to the process by which one makes inferences about a population, based on information obtained from a sample.

How will increasing the confidence level from 95% to 99% affect the confidence interval?

Increasing the confidence will increase the margin of error resulting in a wider interval. A larger margin of error produces a wider confidence interval that is more likely to contain the parameter of interest (increased confidence).

What effects confidence levels?

Interval widens as confidence level increases, therefore, the reverse is true also, that is with decrease in confidence level, confidence interval becomes narrower. Sample size hold inverse relation with confidence interval, that is with increase in sample size, confidence interval becomes narrower. Smaller standard deviation has no influence on confidence interval.

Why is the Standard Error important?

It is used to compute other measures, like confidence intervals and margins of error.

two types of statistical hypotheses

Null Alternative

What describes the uncertainty of sampling method?

The confidence level describes the uncertainty of a sampling method. The statistic and the margin of error define an interval estimate that describes the precision of the method. The interval estimate of a confidence interval is defined by the sample statistic + margin of error.

P-value.

The strength of evidence in support of a null hypothesis is measured by the P-value. Suppose the test statistic is equal to S. The P-value is the probability of observing a test statistic as extreme as S, assuming the null hypothesis is true. If the P-value is less than the significance level, we reject the null hypothesis.

Hypothesis testing Step 1: State the hypothesis

This involves stating the null and alternative hypotheses. The hypotheses are stated in such a way that they are mutually exclusive. That is, if one is true, the other must be false.

Two types of errors can result from a hypothesis test.

Type I Errors Type II Errors

How to interpret a confidence interval

the true population parameter would fall within a range defined by the sample statistic + margin of error 95% of the time.

Confidence intervals are preferred to point estimates, because confidence intervals indicate

(a) the precision of the estimate and (b) the uncertainty of the estimate.

Common confidence levels - need to memorize

90% = + or - 1.645 95% = + or - 1.96 99% = + or - 2.58

Point Estimate = single value, sample mean

A point estimate of a population parameter is a single value of a statistic. For example, the sample mean x is a point estimate of the population mean μ. Similarly, the sample proportion p is a point estimate of the population proportion P.

two-tailed test

A test of a statistical hypothesis, where the region of rejection is on both sides of the sampling distribution

one-tailed test

A test of a statistical hypothesis, where the region of rejection is on only one side of the sampling distribution

Two methods used for statistical inference are

Confidence Intervals Hypothesis Testing

Why use interval estimate (confidence interval) over point estimate (mean)?

Confidence intervals are preferred to point estimates, because confidence intervals indicate (a) the precision of the estimate and (b) the uncertainty of the estimate.

Decision rules

P-value region of acceptance

Forms of Estimation

Point estimation Interval estimation

Hypothesis testing has 4 steps

State the hypotheses. Formulate an analysis plan Analyze Sample data Inerpret the result

hypothesis testing

Statisticians follow a formal process to determine whether to reject a null hypothesis, based on sample data

Why use a confidence interval?

Statisticians use a confidence interval to express the precision and uncertainty associated with a particular sampling method

sampling distribution

Suppose that we draw all possible samples of size n from a given population. Suppose further that we compute a statistic (e.g., a mean, proportion, standard deviation) for each sample. The probability distribution of this statistic is called a sampling distribution.

How to interpret a Margin of Error

Suppose the local newspaper conducts an election survey and reports that the independent candidate will receive 30% of the vote. The newspaper states that the survey had a 5% margin of error and a confidence level of 95%. These findings result in the following confidence interval: We are 95% confident that the independent candidate will receive between 25% and 35% of the vote.

How to interpret a confidence level

Suppose we collected all possible samples from a given population, and computed confidence intervals for each sample. Some confidence intervals would include the true population parameter; others would not. A 95% confidence level means that 95% of the intervals contain the true population parameter; a 90% confidence level means that 90% of the intervals contain the population parameter; and so on.

alternative hypothesis

The alternative hypothesis, denoted by H1 or Ha, is the hypothesis that sample observations are influenced by some non-random cause.

Hypothesis testing Step 2: Formulate an analysis plan.

The analysis plan describes how to use sample data to evaluate the null hypothesis. The evaluation often focuses around a single test statistic.

Which of the following statements is true. I. When the margin of error is small, the confidence level is high. II. When the margin of error is small, the confidence level is low. III. A confidence interval is a type of point estimate. IV. A population mean is an example of a point estimate. (A) I only (B) II only (C) III only (D) IV only. (E) None of the above.

The correct answer is (E). The confidence level is not affected by the margin of error. When the margin of error is small, the confidence level can low or high or anything in between. A confidence interval is a type of interval estimate, not a type of point estimate. A population mean is not an example of a point estimate; a sample mean is an example of a point estimate.

What is the meaning of confidence interval?

The formal meaning of a confidence interval is that X% of the confidence intervals should contain the true population parameter.

Null hypothesis

The null hypothesis, denoted by Ho, is usually the hypothesis that sample observations result purely from chance.

region of acceptance

The region of acceptance is a range of values. If the test statistic falls within the region of acceptance, the null hypothesis is not rejected. The region of acceptance is defined so that the chance of making a Type I error is equal to the significance level. The set of values outside the region of acceptance is called the region of rejection. If the test statistic falls within the region of rejection, the null hypothesis is rejected. In such cases, we say that the hypothesis has been rejected at the α level of significance.

statistical hypothesis

an assumption about a population parameter.

What is the Standard Error?

an estimate of the standard deviation of a statistic.

parameter

any summary number, like an average or percentage, that describes the entire population A parameter is a measurable characteristic of a population, such as a mean or a standard deviation

Statistical inference

drawing conclusions about a population based on a sample

interval estimate (confidence interval)

interval or range of possible values within which a population parameter is likely to be contained

What is a confidence level?

probability value associated with a confidence interval.

Hypothesis testing

refers to the formal procedures used by statisticians to accept or reject statistical hypotheses

"mu" = μ

refers to the mean for a population.

"x bar"

refers to the mean for a sample