Introduction to Inference Statistical Hypotheses Statistical Hypotheses are

Introduction to Inference

Statistical Hypotheses • Statistical Hypotheses are statements about population parameters. • Hypotheses are not necessarily true.

Examples of Hypotheses • mean weight of adult males is greater than 160 pounds • proportion of UK graduate students who have used illegal drugs in the last month is 0. 20 • mean lifetime of GE light bulbs is greater than mean lifetime of Sylvania light bulbs • milk drinkers are better lovers

In statistics, we test one hypothesis against another… • The hypothesis that we want to prove is called the alternative hypothesis, Ha. • Another hypothesis is formed which contradicts Ha. – This hypothesis is called the null hypothesis, Ho contains an equality statement.

Decision • After taking the sample, we must either: Reject Ho and believe Ha or Fail to Reject Ho because there was not sufficient evidence to reject it.

Errors

Probability of Errors • P(Type I Error) = P(Reject Ho | Ho is true) is called the level of significance, . – The smaller is, the harder it is to reject Ho. – We usually choose to be. 05. • P(Type II Error) = P(Fail to Reject Ho | Ho is false). – It is denoted by .

Important Terms • Test Statistic: The statistic we compute to make the decision. Sampling distribution of test statistic given that Ho is true must be known or well approximated. • Critical Region: The values of the test statistic such that we Reject Ho and conclude that Ha is true. – This depends on choice of Ha. • The endpoint of the critical region is called the critical point. – Example: If the critical region is Z > 1. 645 is the critical point.

Steps of a Hypothesis Test • State Ha and Ho • Specify , n , the test statistic and the critical region • Take sample and compute test statistic • Make decision and interpret the results

P-value • The choice of is subjective. • The smaller is, the smaller the critical region. Thus, the harder it is to Reject Ho. • The p-value of a hypothesis test is the smallest value of such that Ho would have been rejected.

Rules • If P-value , reject Ho. • If P-value , do not reject Ho. • Example: p-value = 0. 03 – Reject Ho @ =. 05 – Fail to Reject Ho @ =. 01

Estimation • Population mean, , is an unknown parameter. • Wish to estimate based on a sample. • is a statistic which estimates . • Recall – We call a point estimate because its value is a point on the real line. Unfortunately, if we sample from a continuous distribution, – Thus, we are sure that our estimate is wrong.

Interval Estimates • Statisticians prefer interval estimates. • • Something depends on amount of variability in data and how certain we want to be that we are correct. • The degree of certainty that we are correct is known as the level of confidence. – Common levels are 90%, 95%, and 99%.

Facts • Increasing the level of confidence, – decreases , the probability of error – increases the critical point – widens the interval • Increasing n, decreases the width of the interval