Notes for 3/2/15-3/6/15
Notes for 3/2/15-3/6/15 PSYC 3301
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This 12 page Class Notes was uploaded by Rachel Marte on Saturday March 7, 2015. The Class Notes belongs to PSYC 3301 at University of Houston taught by Dr. Perks in Fall. Since its upload, it has received 125 views. For similar materials see Introduction to Psychological Statistics in Psychlogy at University of Houston.
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Date Created: 03/07/15
3215 The First Two Steps of Hypothesis Testing The Process of Hypothesis Testing As you may recall we follow four steps when performing a hypothesis test 1 State a hypothesis about a population null and alternative hypotheses 2 Use the hypothesis to predict characteristics the sample should have set criteria for a decision on whether or not a sample is significantly different from the population 3 Obtain a random sample from the population 4 Compare the obtained sample to the prediction about the population Step 1 Step I 39 State a hypothesis about a population null and alternative hypotheses The hypotheses 1 The null hypothesis Ho treatment has no effect there is no change or relationship present there is no difference between the population and sample mean they come from the same distribution 2 The alternative hypothesis H1 or Ha treatment has an effect on the dependent variable there is an effect come from a different distribution For each hypothesis test you must state both the null hypothesis AND the alternative hypothesis Each one re ects one of two possible outcomes that there is no difference between the population mean and sample mean or that there is a difference After performing the hypothesis test we will reject one of these hypotheses and retain the other Recall the difference between a onetailed and twotailed hypothesis test Which type of test we are performing affects how you will state the alternative hypothesis As a reminder We use a twotailed test when we simply want to know whether or not there is a difference between the population mean and sample mean but we do not care whether the sample mean is significantly higher or lower than the population mean We can also perform directional hypothesis tests if we want to know what direction the difference is in We use a positive one tailed test if we want to know whether or not the sample mean is significantly higher than the population mean e g Does studying for statistics tests significantly improve students grades We use a negative onetailed test if we want to know whether or not the sample mean is significantly lower than the population mean e g Does drinking signi cantly impair students driving abilities A 5 39 b 1 0 s 39 g i a z A s m 39 pc xifi 3 Hr Llllt39il It39s X39 lii 6 one Lilli d trxi T vc tazlcd CST There are a few different ways you can state the hypotheses You only have to state each one once so pick which way you prefer and stick to it For example if you state the hypotheses in sentence form you do not need to also symbolize them mathematically How to state the hypotheses 1 The null hypothesis 0 Sentence form e g There is no difference between the sample mean and the population mean 0 Symbolization o Hmiu or O Hoip0 2 The alternative hypothesis a Twotailed test 0 Sentence form e g There is a difference between the sample mean and the population mean 0 Symbolization o Huiiu or O H1ip 0 b Positive onetailed test 0 Sentence form e g The sample mean is significantly higher than the population mean 0 Symbolization o Huigtu or 0 H1 2 u gt 0 c Negative onetailed test 0 Sentence form e g The sample mean is significantly lower than the population mean 0 Symbolization o H iltu or 0 H1 2 u lt 0 Step 2 Step 2 Use the hypothesis to predict characteristics the sample should have set criteria for a decision on whether or not a sample is significantly di erent from the population 013 Set the criteria for a decision The data we collect this is part of step 3 but of course we will not actually go through the process of collecting data in classyou will most likely be given relevant information e g sample size population mean and statistics which you will then use to calculate zscores will be used to compare the treatment group to the population We know that the null hypothesis specifies what kind of samples we would get if the treatment had no effect but we need to know what would make the null true Therefore step 2 s purpose is to answer the question Which sample means would indicate that the H0 is true and which sample means would indicate that the H1 is true The distribution of sample means if The null hypothesis is True all the possible outcomes Sample means close To HO highprobability values if H0 is True Extreme low u from H Extreme low probability VOIUGS probability values If HQ is true if HE is True The diagram above indicates which locations on the graph would indicate that we should accept H0 and which indicate that we should reject it for a twotailed test for a onetailed test there would be only one shaded region that would be located on one end of the graph The shaded regions are also called the critical regions and are made up of extremely unlikely values In other words if the sample mean we obtain falls in a critical region we should reject H0 and accept H1 If the sample mean falls in the unshaded region we should accept H0 and reject H A value in the critical region would be very hard to obtain if Hg was true ie it would be di icult to get a mean with such a low probability if there really is no di erence between the sample mean and the population mean which is why we would reject H0 In order to decide which sample means would indicate that the H0 is true we choose an alpha 0t level The alpha level informs us about high probability and low probability samples and is also referred to as the level of significance The most commonly used alpha levels are 05 01 and 001 when in doubt assume 0t05 since it is the most common alpha level An alpha level of 05 0t05 indicates that we will only reject Ho if the sample falls into the 5 most extreme sample means The alpha level we choose will determine the critical region There is a difference between the alpha level and the critical region The alpha level is a probability value that is used to define the very unlikely sample outcomes if the null hypothesis is true The critical region is composed of the samples values that are unlikely if the Ho holds true If sample data fall in the critical region then we reject the null hypothesis In other words an alpha level of 05 means that the critical region contains 5 of the most extreme scores in the distribution The alpha level is just the probability value we use to mark off the critical region and the critical region is the actual area of the graph made up of the extremely unlikely sample mean values we would use to reject the null hypothesis For your alpha level of choice you need to find the boundaries that separate your critical region from the rest of the distribution this will be a zscore One important point to keep in mind is that if you are performing a twotailed test you must divide your critical region in half For example if 0t05 and you are performing a twotailed test you need have a critical region at each extreme end of the graph that begin at a probability of 025 In order to determine the boundary of the critical regions you must use the unit normal table to find the zscore that matches your alpha level Look for the probability closest to your alpha level in column c after all the critical regions are always in the tails of the distribution since it covers the extreme values of the distribution If you can t nd the exact probability in column C pick the closest one to your alpha level and if two probabilities are equally close choose the more conservative value lower zscore which would result in a larger critical region The zscore that matches that probability marks the beginning of the critical regions and is denoted Zcrit It might be helpful if you memorize the Zcrit values that match each alpha level as they do not change If you do not remember them you can always look them up in the unit normal table Below I will show you how to determine the Zcrit values for the most common alpha levels A 3 c 1 1 949 1515 4495 1a5 95 5 I U455 I L 9515 1435 il The closest probability to 05 in column C is 0495 The corresponding zscore in column A is 165 Therefore zenF1 65 for a 05 If you are performing a positive onetailed hypothesis test zcm165 and if you are performing a negative onetailed test zcm165 This is illustrated below a 05 positive onetailed test H1 6 gt H O 165 a 05 negative onetailed test Ehi ltll 165 0 If you are performing a twotailed test with OF 05 you must divide alpha in half resulting in two critical regions defined by OF 025 Find the zscore corresponding to this probability in column C of the unit normal table A B C D 5 HEP14 gm L QT L125g 45 L9 Elj E 4 4EEEJ The closest probability to 025 in column C is 0250 The corresponding zscore in column A is 196 Therefore zent i 196 for a025 Note that there are two critical regions This is illustrated below a 05 so a 025 twotailed test H1 3 75 11 p025 196 O 196 A B C D 999 9999 9 9999 I999 9999 999 9999 9999 The closest probability to 01 in column C is 0099 The corresponding zscore in column A is 233 Therefore zcrit233 for a 01 If you are performing a positive onetailed hypothesis test zcm233 and if you are performing a negative onetailed test zcm233 This is can be illustrated in the same way as OF 05 is illustrated above If you are performing a twotailed test with OF 01 you must divide alpha in half resulting in two critical regions defined by OF 005 Find the zscore corresponding to this probability in column C of the unit normal table A B C D 9999 99 9999 999 9991 9991 999 9999 9999 The closest probability to 005 in column C is 0049 The corresponding zscore in column A is 258 Therefore zent i 258 for a005 Note that there are two critical regions This can be illustrated in the same way as OF 025 is illustrated above You can calculate the Zcrit value for any other alpha level in the same way as we have done in the two examples above 3415 The Last Two Steps of Hypothesis Testing Step 3 Step 3 Obtain a random sample from the population 0R Collect data and compute sample statistics After we have done all the prep work stating the hypotheses alpha level and Zcrit values we must collect data Of course we will not actually go through the process of collecting data in class Most likely you will be given any relevant information e g sample size population mean and statistics which you will then use to calculate zscores However if you were actually performing an entire hypothesis test in the real world this is where you would obtain relevant sample data or summarize sample data to get sample statistics Once you have collectedbeen given the datastatistics you will use that information to calculate the test statistic As of now we are using zscores as our test statistic of choice Therefore we need the sample mean because we want to compare it to the population mean Since we are interested in comparing sample means to population means we will use the formula for zscores for sample means the one that includes the standard error Remember that the zscore formula describes a ratio of the obtained difference versus the difference due to chance 2 oxh Note Recall that E is the sample mean u is the population mean a is the population standard Formula for zscores for sample means 2 deviation and n is the sample size oxn is the formula for the standard error The zscore you calculate is often referred to as zobt the obtained zscore or zcalc the calculated zscore This just means that you calculated this zscore from the sample statistics instead of looking it up in the unit normal table like Zcrit Step 4 Step 4 Compare the obtained sample to the prediction about the population 0R Make a decision After getting zobt in step 3 you need to use it to decide whether to reject or retain the null hypothesis in accordance to the criteria you established in step 2 If zobt falls within the critical region you will reject Ho and accept H1 If zobt does not fall within the critical region you will retain Ho and reject H1 The easiest way to determine whether or not zobt is in the critical region is to sketch a graph where you label the population mean 20 the ZcritV 111 S and shade in the critical regions and the zobt value If zobt falls in a shaded region reject the null Putting it All Together Example 1 You are interested in assessing if students who get a good night s sleep learn more information in class the following day than the normal student To test this you collect sample data from a group of 25 students who slept well and find that their average learning score is 40 If the population mean learning score is 30 with a standard deviation of 10 can you conclude that people who get a good night s sleep learn more than normal students List information 30 i40 610 n25 assume 105 since the alpha level isn t specified positive onetailed test Do people who sleep well learn more than those who don t Step I 39 State a hypothesis about a population null and alternative hypotheses H0 Students who sleep well do not learn better than those who do not i u H1 Students who sleep well learn better than those who do not i gt u Step 2 Set the criteria for a decision For a positive onetailed test with 105 Zcrit165 find Zcrit using the unit normal table See page 5 for how I found this value for an alpha level of 05 Step 3 Collect data and compute sample statistics z 33 Choose the proper equation 2 1437215 Plug in known quantities z 2 gs Subtract 30 from 40 and take square root of 25 z 12 0 Divide 10 by 5 in denominator zobt 5 Simplify Step 4 Make a decision LL 0 1 65 5 Zcrit Zobt Because zobt falls within the critical region we will say that there is a significant difference We will reject Ho Our data suggest that students who sleep well learn better than those who do not Example 2 While driving to class you start to think that the lane you are in is the slowest You want to know if the lane you are in middle is slower than the other lanes The population data shows that the average speed is 50mph with a standard deviation of 12 You collect a sample 36 cars and find that in your lane the average speed is only 45 Can you conclude that your lane is statistically slower than the other lanes List information Step I Step 2 Step 3 Step 4 u50 i45 612 n36 assume 0t05 since the alpha level isn t speci ed negative onetailed test Is your lane slower than the other lanes State a hypothesis about a population null and alternative hypotheses H0 Your lane does not go at a different speed than the other lanes 2 u H1 Your lane goes slower than the other lanes 2 lt u Set the criteria for a decision For a negative onetailed test with 0t05 Zcrit 165 find Zcrit using the unit normal table See page 5 for how I found this value for an alpha level of 05 Note that zc t is negative because this is a negative onetailed test we re only interested in if your lane is slower than the other lanes not whether or not it s faster or slower Collect data and compute sample statistics Z p z CINE Choose the proper equation 45 50 z 12 m Plug in known quantities 56 Subtract 50 from 45 and take square root of 36 z is Divide 12 by 6 in denominator Zobt 3925 Make a decision p05 5 165 O Zobt Zcrit Because zobt falls within the critical region we will say that there is a significant difference We will reject Ho Our data suggest that your lane does go slower than the other lanes 3615 Hypothesis Testing In uencing Factors Significance Effect Size and Error In uencing Factors There are several factors that can in uence the outcomes of hypothesis tests 0 The size of the difference between the sample mean i and the population mean u o This value appears in the numerator of the zscore formula Z u o This is the factor we want to in uence the hypothesis test after all the point of the hypothesis test is to determine if there is a significant difference between the sample mean and population mean 0 The variability of the scores 0 The variability in uences the size of the standard error in the denominator of the zscore formula ax o The variability is measured by either the standard deviation or the variance 0 The number of scores in the sample n o The sample size 11 also in uences the size of the standard error in the denominator of the zscore formula ax o The bigger the sample size the more likely you will end up rejecting Ho These different factors minus the first one can make using hypothesis testing a little unreliable Whether or not you reject the null could depend on how big your sample size or how variable the population is However despite some talk about finding another method to replace it hypothesis testing doesn t seem to be going anywhere anytime soon Statistical Signi cance A result is statistically significant if it is very unlikely to occur when the null hypothesis is true In other words a result is statistically significant if 0 It is sufficient to reject Ho ie the decision from the hypothesis test is to reject Ho 0 It is greater or less than would be expected by chance Effect Size Effect Size is a way of quantifying the difference between two groups that has many advantages over the use of tests of statistical significance alone Effect size emphasizes the size of the difference rather than confounding this with sample size Hypothesis testing tells you if the result is significant or not but it doesn t tell you about the size of an effect In other words the effect size measures the strength of a relationship instead of just saying whether or not a result is significant it tells you how significant it is and removes the variability of sample size from the equation Effect sizes are often reported along with the results of a hypothesis test We will calculate the effect size using Cohen s d mean difference standard deviation Cohen s d Size of the effect 0 Small effect d i 2 0 Medium effect d i 5 0 Large effect d i 8 In the real world the numbers you get for Cohen s d will probably be relatively small but theoretically e g in this class they could be bigger Don t freak out if you get a big effect size in class There are other ways to calculate effect size that other statisticians have come up with but this is a common formula The scale above has a lot of gray areas in it Is an effect size of 3 small or medium Either pick one or the other if the context of the problem suggests one over the other or just say that it s a small to medium effect You only really need to calculate the effect size if you reject Ho After all it tells you how significantlarge an effect is If you retain Ho you re saying that there isn t a difference to begin with so why would you try to calculate how significant a difference is Example After running a hypothesis test you found that students who binge drink have significantly lower attendance than most students For the study the population mean attendance is 80 with a standard deviation of 15 Your sample of 9 students who binge drink had an average attendance of only 50 Using Cohen s d calculate the size of this effect Cohen s d Choose the proper equation d 501 580 Plug in known quantities d 1 350 Subtract 80 from 50 d 2 Simplify Notice Even though we were given n in this problem we didn t need to use it E ect size does not depend on the sample size Error There are two types of errors we can make in hypothesis testing 0 Type I Error When the data lead you to reject Ho even though it is in fact true 0 In other words a type I error occurs when you conclude that a treatment has an effect when it actually does not have an effect I zobt falls in the critical region even though it is not actually significant I The smaller the alpha level is the smaller the critical region is the less likely zobt is to fall inside the critical region the less likely a type I error is Hypothesis tests are designed to minimize type I error This can also be called a false positive 0 Type I error is equivalent in size to the size of the critical region e g if 105 we will incorrectly reject Ho about 5 of the time I The smaller the alpha level the less likely a type I error is 0 Type II Error When the data lead you to retain Ho even though it is false 0 In other words a type I error occurs when you conclude that a treatment has no effect when it actually does have an effect I zobt does not fall in the critical region even though it is actually significant I The bigger the alpha level is the bigger the critical region is the less likely zobt is to fall outside of the critical region the less likely a type II error is o This can also be called a false negative 0 The bigger the alpha level the less likely a type II error is As you can see the alpha level affects the chances of both a type I and a type II error If we choose a smaller alpha we will be less likely to get a type I error but we will be more likely to get a type II error If we choose a bigger alpha we will be less likely to get a type II error but we will be more likely to get a type I error In other words we need to nd a balance we can t make the alpha level either too big or too small
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