# Anthony Greene1 Simple Hypothesis Testing Detecting Statistical Differences In The Simplest Case:  and  are both known I The Logic of Hypothesis Testing:

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Anthony Greene1 Simple Hypothesis Testing Detecting Statistical Differences In The Simplest Case:  and  are both known I The Logic of Hypothesis Testing: The Null Hypothesis IIThe Tail Region, Critical Values: α IIIType I and Type II Error

Anthony Greene2 The Fundamental Idea 1.Apply a treatment to a sample 2.Measure the sample mean (this means using a sampling distribution) after the treatment and compare it to the original mean 3.Remembering differences always exist due to chance, figure out the odds that your experimental difference is due to chance. 4.If its too unlikely that chance was the reason for the difference, conclude that you have an effect

Anthony Greene3 Null and Alternative Hypotheses Null hypothesis: A hypothesis to be tested. We use the symbol H 0 to represent the null hypothesis. Alternative hypothesis: A hypothesis to be considered as an alternate to the null hypothesis. We use the symbol H a to represent the alternative hypothesis.

Anthony Greene4 The Distribution of Sample Means As The Basis for Hypothesis Testing The set of potential samples is divided into those that are likely to be obtained and those that are very unlikely if the null hypothesis is true.

Anthony Greene5 The Logic of the Hypothesis Test 1.We start with knowledge about the distribution given no effect (e.g., known parameters or a control group) and the data for a particular experimental treatment 2.Begin with the assumption that there is no experimental effect: this is the null hypothesis 3.Compute the probability of the observed data given the null hypothesis 4.If this probability is less than  (usually 0.05) then reject the null hypothesis and accept the alternative hypothesis

Anthony Greene6 The Logic of the Hypothesis Test The critical region (unlikely outcomes) for  =.05.

Anthony Greene7 Avoiding Confusion About z z crit vs. z obs z crit z obs

Anthony Greene8 Air Puff to Eyeblink Latency (ms)

Anthony Greene9 95% of all samples of 25 eyeblinks have mean within 1.96 standard deviations of 

Anthony Greene10 Probability that the sample mean of 450 ms is a chance difference from the null- hypothesis mean of 454 ms z = -2.56

Anthony Greene11 Using More Extreme Critical Values The locations of the critical region boundaries for three different levels of significance:  =.05,  =.01, and  =.001.

Anthony Greene12 Test Statistic, Rejection Region, Nonrejection Region, Critical Values

Anthony Greene13 Rejection regions for two- tailed, left-tailed, and right- tailed tests While one-tailed tests are mathematically justified, they are rarely used in the experimental literature

Anthony Greene14 Graphical display of rejection regions for two-tailed, left- tailed, and right-tailed tests

Anthony Greene15 α for 1 and 2-tailed tests α/2 αα

Anthony Greene16 α for 1 and 2-tailed tests for α = 0.05

Anthony Greene17 Correct and incorrect decisions for a hypothesis test

Anthony Greene18 Correct and incorrect decisions for a hypothesis test + = + = 1.00

Anthony Greene19 Type I and Type II Errors Type I error: Rejecting the null hypothesis when it is in fact true. Type II error: Not rejecting the null hypothesis when it is in fact false.

Anthony Greene20 Significance Level The probability of making a Type I error, that is, of rejecting a true null hypothesis, is called the significance level, , of a hypothesis test. That is, given the null hypothesis, if the liklihood of the observed data is small, (less than  ) we reject the null hypothesis. However, by rejecting it, there is still an  (e.g., 0.05) probability that rejecting the null hypothesis was the incorrect decision.

Anthony Greene21 Relation Between Type I and Type II Error Probabilities For a fixed sample size, the smaller we specify the significance level, , (i.e., lower probability of type I error) the larger will be the probability, , of not rejecting a false null hypothesis. Another way to say this is that the lower we set the significance, the harder it is to detect a true experimental effect.

Anthony Greene22 Possible Conclusions for a Hypothesis Test If the null hypothesis is rejected, we conclude that the alternative hypothesis is true. If the null hypothesis is not rejected, we conclude that the data do not provide sufficient evidence to support the alternative hypothesis.

Anthony Greene23 Critical Values, α = P(type I error) Suppose a hypothesis test is to be performed at a specified significance level, . Then the critical value(s) must be chosen so that if the null hypothesis is true, the probability is equal to  that the test statistic will fall in the rejection region.

Anthony Greene24 Some important values of z  

25 Power The power of a hypothesis test is the probability of not making a Type II error, that is, the probability of rejecting a false null hypothesis. We have Power = 1 – P(Type II error) = 1 –  The power of a hypothesis test is between 0 and 1 and measures the ability of the hypothesis test to detect a false null hypothesis. If the power is near 0, the hypothesis test is not very good at detecting a false null hypothesis; if the power is near 1, the hypothesis test is extremely good at detecting a false null hypothesis. For a fixed significance level, increasing the sample size increases the power.

Anthony Greene26 Basic Idea H 0 : Parent distribution for your sample if there IS NO effect μ 0 = 40 Z crit = 1.64 M = 42 Fail to reject M = 48 Reject null hypothesis Conclude Effect

Anthony Greene27 Basic Idea H 0 : Parent distribution for your sample if there IS NO effect H a : Parent distribution for your sample if there IS an effect μ 0 = 40 μ a = ? Z crit = 1.64

Anthony Greene28 Basic Idea H 0 : Parent distribution for your sample if there IS NO effect μ 0 = 40 Z crit = 1.64 α 1 - α

Anthony Greene29 Basic Idea H a : Parent distribution for your sample if there IS an effect μ a = ? Z crit = 1.64 β 1 - β

Anthony Greene30 Basic Idea H 0 : Parent distribution for your sample if there IS NO effect H a : Parent distribution for your sample if there IS an effect μ 0 = 40 μ a = ? Z crit = 1.28 We can move z crit Z crit = 2.58

Anthony Greene31 Basic Idea H 0 : Parent distribution for your sample if there IS NO effect H a : Parent distribution for your sample if there IS an effect μ 0 = 40 μ a = ? Z crit = 1.64 We can increase n

Anthony Greene32 The one-sample z-test for a population mean (Slide 1 of 3) Step 1The null hypothesis is H 0 :  =  0 and the alternative hypothesis is one of the following: H a :    0 H a :   0 (Two Tailed)(Left Tailed)(Right Tailed) Step 2Decide on the significance level,  Step 3The critical values are ±z  /2 -z  +z  (Two Tailed)(Left Tailed)(Right Tailed)

Anthony Greene33 The one-sample z-test for a population mean (Slide 2 of 3) α/2 αα

Anthony Greene34 The one-sample z-test for a population mean (Slide 3 of 3) Step 4Compute the value of the test statistic Step 5If the value of the test statistic falls in the rejection region, reject H 0, otherwise do not reject H 0.

Anthony Greene35 Synopsis

Anthony Greene36 P-Value To obtain the P-value of a hypothesis test, we compute, assuming the null hypothesis is true, the probability of observing a value of the test statistic as extreme or more extreme than that observed. By “extreme” we mean “far from what we would expect to observe if the null hypothesis were true.” We use the letter P to denote the P-value. The P-value is also referred to as the observed significance level or the probability value.

Anthony Greene37 P-value for a z-test Two-tailed test: The P-value is the probability of observing a value of the test statistic z at least as large in magnitude as the value actually observed, which is the area under the standard normal curve that lies outside the interval from –|z 0 | to |z 0 |, Left-tailed test: The P-value is the probability of observing a value of the test statistic z as small as or smaller than the value actually observed, which is the area under the standard normal curve that lies to the left of z 0, Right-tailed test: The P-value is the probability of observing a value of the test statistic z as large as or larger than the value actually observed, which is the area under the standard normal curve that lies to the right of z 0,

Anthony Greene38 Guidelines for using the P-value to assess the evidence against the null hypothesis

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