Estimation Making Educated Guesses Point Estimation Interval Estimation

Estimation: Making Educated Guesses • Point Estimation • Interval Estimation • Hypothesis Testing

Case Ia • Does a particular sample of observations in this study come from a specified population or does it represent a different population? – “Known” population mean – “Known” population standard deviation

The 4 th Grade Case • Suppose you are the superintendent of schools and you discover that the average reading achievement of your 4 th graders has fallen far below that of previous years. One explanation posed by the teachers is that the district is faced with an unusually dull group of 4 th graders this year. The teachers suggest that the average verbal IQ of this year’s 4 th graders is far different from the national average and that is why reading achievement is so low. • You know that IQ-test scores don’t change much from year to year unless a school system is affected by changes in its attendance (e. g. a large migration of new families). Your school system has remained quite stable, but you decide to check out the teacher’s claim • You have a limited budget and while you have extensive achievement data on the 4 th graders, you have limited IQ data available. So you decide to test a sample of 400 4 th graders rather than all 5000 of them.

The Logic of Hypothesis Testing • Null hypotheses (H 0) • Alternative hypotheses (H 1) • Is this backwards and convoluted or what?

Hypothesis Testing: General Model • Identify the population and population parameter of interest • Define the null hypothesis and alternative hypothesis • Collect data on a random sample selected from population of interest • Compute a sample statistic that is an estimate of the parameter of interest • Decide on a criteria for evaluating the sample evidence • Make decision to retain the null hypothesis or discard the null hypothesis in favor of the alternative hypothesis

Error and Risk

Type I Error and Level of Significance • Type I error: the mistake of rejecting the null hypothesis (H 0) when in fact it is true. • Level of Significance: – Alpha ( ) =. 05 – Significant at the. 05 level – p <. 05

Type II Error • Type II Error: If the alternative hypothesis (HA) is true and the decision maker decides to stick with the null hypothesis (H 0) • Risk

Hypothesis Testing: General Model • Identify the population and population parameter of interest • Define the null hypothesis and alternative hypothesis • Collect data on a random sample selected from population of interest • Compute a sample statistic that is an estimate of the parameter of interest • Decide on a criteria for evaluating the sample evidence • Make decision to retain the null hypothesis or discard the null hypothesis in favor of the alternative hypothesis

Decision Rules • Decision Rule: the values of sample statistic that keep you believing H 0 and the values that lead you to reject H 0

Hypothetical Frequency Distribution of 1000 Samples. 3413 . 1359 68% . 0214 95% . 0013 97. 75 . 1359 . 0013 99% 98. 5 99. 25 100 = population mean 100. 75 101. 5 102. 25

“How Likely? ” • How likely is this sample mean to arise by sampling error? • The “Sampling Distribution of Means” provides a model of what to expect if the null hypothesis is true Likely IQ = 100 Population of Scores Unlikely IQ = 100 Sampling Distribution of Means • By convention, an unlikely sample mean under the null hypothesis occurs 5 in 100 times (. 05) or 1 in 100 times (. 01)

Selecting a Level of Significance: What is Unlikely? • Goal is to determine how consistent or inconsistent the sample data are with the null hypothesis • Usually select some small (conservative) level of significance (. 05, . 01, . 001) • Level chosen depends on seriousness of the consequences of one’s decision

Hypothetical Frequency Distribution of 1000 Samples 68% Unlikely at. 05 Unlikely at. 01 95% 99% 97. 75 98. 5 99. 25 100 population mean 100. 75 101. 5 102. 25

One- and Two-Tail Test? • One- and two-tail tests tell you which tail(s) in the sampling distribution of means should be used to determine “How likely? ” • Two-Tail Test: Willing to Entertain a Sample Mean in Either Tail--H 1 : Population Mean not = 100 • One-Tail Test: Willing to Specify the Direction of the Sample Mean (Above or Below the Population Mean Under the Null Hypothesis): H 1 : Population Mean > 100 Two Tail . 025 One Tail . 025 . 05

Critical Values for Case Ia: Z-Test

Sampling Distribution of Means: Standard Errors, Critical Values, and Ps Z Distribution Normal Curve Two tailed Test -2 se Critical Values P= -2. 58 se -1. 96 se . 05 = outisde of 1. 96 either end u +1 se +2 se +1. 96 se . 01 = outside of 2. 58 either end +2. 58 se

Sampling Distribution of Means: Standard Errors, Critical Values, and Ps Z Distribution Normal Curve -2 se Critical Values P= One tailed test -1 se u +1 se +2 se +1. . 65 se. 05 +2. 33 se. 01

Sampling Distribution of Means: Standard Errors, Critical Values, and Ps Z Distribution Normal Curve One tailed test -2 se -1 se Critical Values -2. 33 se -1. . 65 se P= . 01 . 05 u +1 se +2 se

The Decision Regarding H 0: The Lingo • Reject H 0 : Take position that null hypothesis is probably false – – “H 0 (the null hypothesis) was rejected” “A statistically significant finding was obtained” “A reliable difference was observed” “p is less than X” (a small decimal value (p<. 05, p<. 01)) • Fail-to-reject H 0: Take the position that there is not enough evidence to reject the null hypothesis – “H 0 was tenable” – “H 0 was accepted” – “No reliable differences were observed” – “No significant differences were found” (ns) – “p is greater than X” (a small decimal value (p>. 05, p>. 01))

Significance Testing vs Hypothesis Testing • Hypothesis Testing: – Alpha level is preset – Decision is “reject” or “do not reject” – Don’t discuss impressive p-levels • Significance Testing – No alpha levels preset – Data speak through p-levels – Strength of significance discussed