Notes for 2/16/15-2/18/15
Notes for 2/16/15-2/18/15 PSYC 3301
Popular in Introduction to Psychological Statistics
verified elite notetaker
Popular in Psychlogy
verified elite notetaker
PSYC 3310 Industrial-Organizational Psychology
verified elite notetaker
This 12 page Class Notes was uploaded by Rachel Marte on Wednesday February 25, 2015. The Class Notes belongs to PSYC 3301 at University of Houston taught by Dr. Perks in Fall. Since its upload, it has received 123 views. For similar materials see Introduction to Psychological Statistics in Psychlogy at University of Houston.
Reviews for Notes for 2/16/15-2/18/15
Report this Material
What is Karma?
Karma is the currency of StudySoup.
You can buy or earn more Karma at anytime and redeem it for class notes, study guides, flashcards, and more!
Date Created: 02/25/15
2 l 6 15 ZScores and Probability Probability Basics Probability refers to the likelihood of obtaining a certain outcome All inferential procedures are built around the idea of probability and we usually start With a population and use probabilities to describe possible samples INFERENTIAL STATISTICS Population Somple PROBABILITY The diagram shows which processes we use when trying to describe either populations or samples If we start out with a sample we use it to determine statistics and then use inferential statistics to describe the population If we start out with a population we use it to determine probabilities and then use probability to describe possible samples of total oucomes classified as A The probability of A C 3 P We symbolize probability With the notation total of possible outcomes Probability and the Normal Distribution Different portions or areas of graphs represent different probabilities The total area under a curve is 1 so all the different areas under the curve must have probabilities that add up to l or 100We usually use decimals rather than percentages in statistics The diagram above shows the probabilities of di erent areas under the normal curve As you should remember measurements for the normal curve are made in increments of standard deviation and the percentages above re ect the percentage of scores that should fall into each category For instance about 68 of data fall within one standard deviation on either side of the mean 95 of data fall in the ZSD to ZSD range and 997 of data fall in the 3SD to 3SD range Therefore only a very small percentage of scores will fall outside of 3 SDs of the mean This means that the probability of obtaining a score that is between the mean z0 and a z score of I should be about 341 or 341 Obtaining a score between I and 1 standard deviation has a probability of approximately 682 Recall that normal distributions like the one pictured above are symmetrical with the highest frequency in the middle and ends tapering off The normal shape can be described mathematically or by the proportions of area contained in each section of the distribution as shown in the diagram above Locations are identified by zscores so proportions will apply to any normal distribution regardless of its mean and standard deviation Finding proportions for normally distributed variables 1 Find the zscores 2 Draw a graph with the zscores and mean labeled 3 Look up the probability associated with that place on the curve ie zscore using the unit normal table The Unit Normal Table The unit normal table also called the standard normal table or Z table lists the values of the cumulative distribution function of the normal distribution We use it to find the probability that a statistic is observed below above or between values on the standard normal distribution The probability of obtaining any one particular value is essentially zero so the only meaningful questions about probability are those that ask for the probability of obtaining a score that falls within a certain range Once you have determined a zscore and want to know the probability of obtaining a score above or below it or between z and the mean or another zscore it is easy to locate the zscore in question on the table A series of probabilities will be listed next to the zscore each corresponding to a different area of the normal curve in relation to the zscore It also possible to be given a probability and use it to find the corresponding zscore All you need to do is locate the correct probability in the table and see which zscore it is listed under Below is an excerpt of the unit normal table TilELLE 31 THE LiNllT NORMAL T ELEt E lumn A lista asmrL whiz5 A vertical line draian through a manual distribution at a zseure Mention divide the distribution mm two t li Column B identi es the prup ritiun in 111 larger Section culil iiht their Column C identi es the pruporitinn in the smaller Eectiom called the mil Column D identi es the proportion between the mean and the aseme Nut2quot Became the normal distributiun is symmetrical the pmpurtiuns for negatiw a wma are lli S m as Klimt for positive grooms 33239 W 53 in F39s EH EC ED Proportion ngmriiun Prepmien Brownian FWi ii 39Prmannim z in Emir in Tail BEAMum Mean and z I 2 in Ensign ill Tail Eelwmn Mam Ii39IJ mm mm 39 p15 593 wij 1393 ELM EEHIEI J l 115 4325 3193 1925 39J E E 4931 U Eli 3935 105439 033 5111 4331 Uli39l 323 l i 335 3 13 094 5 i l 513449 31 I543 329 5 1411 3359 1 1411 39 5199 4513 5195quot Bill 511quot 3321 1T9 U 523939 ATE DEE5 39 Eli El 52 397quot ETEE 1217quot EHEITI 52 ATE EJE TE L32 5255 3T45 1155 133 5319 4531 EE l9 H313 6293 BTW 1293 33393 5359 4 541 335 LE4 631 315 lial 393 l I 5393 il U39E 3393 3135 3632 l As you can see the unit normal table is divided into four columns The first A is the zscore The zscores on the table generally range from 0 to 4 and count by 01 Note that only positive 2 scores are listed If you have a negative zscore you take the absolute value and use the probability listed for its positive counterpart The other three columns list different probabilities associated with zscores in column A In order to understand the differences between the probabilities listed in the other columns we must first distinguish between the body and tail of a distribution If we draw a vertical line through the curve at any random point we divide the distribution into two parts the body and the tail The body of the distribution is the part of the distribution that contains the mean Therefore if we drew a line at a zscore of 2 the body of the distribution would range from a zscore of 3 to 2 because the mean of the distribution z 0 falls within that portion of the curve this is assuming cutoff points of 3 and 3 SDs The tail of the distribution is the part of the distribution that does not contain the mean Using the example above the tail would range from a zscore of 2 to 3 because the mean does not fall in this range 0 is less than 2 so obviously falls in the other part of the graph Using the position of E on the graphs above it is easy to determine which part of the distribution is the body B and which is the tail T The body always includes 9 Column B lists the proportion in the body of the distribution If you are looking for everything either above or below the zscore and that area happens to include the mean therefore making it the body of the distribution then you use the proportion listed in column B At the top of the unit normal table three graphs are displayed The first two graphs show situations in which you would use column B because the shaded regions include the mean If you need to find the probability of the shaded region on either of the graphs above you would use column B because the shaded regions include the mean Column C lists the proportion in the tail of the distribution If you are looking for everything either above or below the zscore and that area does not include the mean therefore making it the tail of the distribution then you use the proportion listed in column C If you look at the graphs at the top of the unit normal table again you would notice that none of them correspond to this situation However if you were looking for the area of the unshaded regions of graphs one and two you would use column C If you need to find the probability of the shaded region on either of the graphs above you would use column C because the shaded regions do not include the mean Notice that these graphs are identical to the ones used as examples above for column B The only di erence is the area of the curve shaded If you are interested in a zscore of 2 you might have to use either column B or column C depending on whether you are looking for the probability of getting a score that is below or above that z 2 Column D lists the proportion between the relevant zscore and the mean The third graph at the top of the unit normal table shows a situation in which you would use column D because the shaded area is between 2 and the mean If you need to find the probability of the shaded region on either of the graphs above you would use column D because the shaded regions are between the mean and the zscore Example 1 For intelligence the population mean is 100 u and the standard deviation is 15 6 You draw these samples at UH 112 for engineering x3 108 for psychology xp and 121 for physics xph How likely are you to obtain these or higher values based on probability under the normal curve Step I Find the zscore s xu 039 a Z 112 100 12 8 39 E 15 15 39 108 100 8 b 2 P 15 15 53 121 100 21 C th 15 1 5 Step 2 Draw a graph with the zscore s and mean labeled a Step 3 Look up the probability associated with that place on the curve ie zscore using the unit normal table a A B C D 1119 1115 214 2135 luau 1331 I 2119 I 2331 I 1131 1911 1190 9111 pxE 2 112 2119I Note We will use column C to nd all three of these probabilities because we are looking for the area of the distribution s tail See the graphs in Step 2 for a visual aid b A B C D 1151 5935 31115 1935 11511 111151 I 2931 I 21119 1154 312154 2946 21154 pxP 2 108 2981I c A B C D 139 911 0323 am 141 9111 mag Aim Ipxp 2 121 0808 Example 2 What zscore separates the top 10 from the bottom 90 Note This problem requires you to find the probability in order to find the zscore which is the opposite of what we have been doing thus far The wording of these questions is very important so that you know which column to take the probability out of For instance if you want to know what score separates the top 15 from the bottom 85 you need to find the probability in column C closest to 15 because the zscore is obviously above the mean so the probability of getting the zscore or higher requires finding the area of the tail or in column B closest to 85 because the zscore is obviously above the mean so the probability of getting the zscore or lower requires finding the area of the body Determining which column to use is much easier if you draw out a graph for the problem There are two basic ways to solve this problem Either you can look for the zscore corresponding to the probability of 1 or above ie the top 10 or you can look for the zscore corresponding to the probability of 9 or below ie the lower 90 Half of the distribution has a probability of 5 since the total area under the curve is 1 the area under half the curve is 5 so z must be located above the mean That is really all the information you need in order to draw a line on the graph but if you want to me more accurate you know that only 25 of data lies above a zscore of 2 so a zscore for 10 must lie somewhere between z1 and z2 The first graph above seeks to solve the problem in the first way mentioned Now that we have established that z is above the mean we know that if we want to find what z is for a probability of 1 or higher we must shade the area above z This area does not include the mean so we are looking for the area of the tail and need to use column C We would look for the closest probability to 1 in column C you might not find 1 exactly just find the number that is closest to 1 and find what the corresponding zscore is The second graph above seeks to solve the problem in the second way mentioned Now that we have established that z is above the mean we know that if we want to find what z is for a probability of 9 or lower we must shade the area below z This area includes the mean so we are looking for the area of the body and need to use column B We would look for the closest probability to 9 in column C and find what the corresponding zscore is Way 1 Look for p 1 in column C and find matching zscore A B C D 121 11111111 111211 3931 F13 5991 113 3119 1111 311151 111935 111 15 Way 2 Look for p 9 in column B and find matching zscore A B C D 111 11111111 1111sz 13111111 I123 I LEQWI 1 3991 I 1111 311153 11193572 111115 Example 3 A sample has a mean 2 of 36 and sd of 4 What proportion is located between the raw scores of 30 x1 and 38 x2 Step I Find the zscore s x i SD 30 36 6 a 21 3915 4 4 38 36 2 b Zz 4 4 Step 2 Draw a graph with the zscore s and mean labeled p1 P2 YI 191 192 Step 3 Look up the probability associated with that place on the curve ie zscore using the unit normal table Note We will be using column D on the table so that we can find p1 and p2 and add them together to get p We will find both probabilities by finding the probability between the mean and each of the two 2 s so we must use column D a Find p 15ltXlt0 or p1 Note Negative zscores do not appear on the unit normal table Take the absolute value of I5 to get 15 and use that to find the proper probability A B C D 1 49 9319 0631 43 3 if Inn 533 I 94332 I L51 3315 i 4345 I Find p 0ltXlt5 or p2 A B C D my my 3125 gum I 5 591 5 3335 Jamil ELSE man i 1950 p1 192 4332 1915 6247 21815 Probabilities and Samples Distribution of Sample Means Sampling Distributions Up to this point we have used zscores to represent individual scores but they can also represent the position of entire samples in a population of scores Since samples are variable two samples are likely to be different even if they were taken from the same population You can theoretically get thousands of different samples from a population However even though none of the samples are exactly alike we can still establish rules for the behavior of all the samples that make up a population The distribution of sample means is the collection of sample for all the possible random samples of a particular size n that can be obtained from a population Distributions of sample means are not based on raw scores but on means this is why zscores will now represent entire samples instead of individual scores Sampling distributions are distributions of statistics that are obtained when you select all possible samples of a specific size from a population The sampling distribution centers on the population mean 11 while different sample means i are turned into zscores that make up the distribution Sample means will be normally distributed An example of creating a sampling distribution If you were creating a sampling distribution for the scores 3 5 7 and 9 you would have to take all the samples you can given a certain sample size For example you could create all the possible samples of n2 In that case there would be 16 possible samples You would then have to find the mean for each sample and then use them to make the distribution Statisticians usually use computers for this process because as you can see it can become very time consuming Central Limit Theorem The central limit theorem CLT states that the distribution of the sum or average of a large number of independent identically distributed variables will be approximately normal regardless of the underlying distribution In other words the sampling distribution of any statistic will be normal or nearly normal as long as the sample size is large enough The distribution of sample means approaches normality when the sample size is around 30 For any population with a mean and a standard deviation the sample means for samples of size n will have a mean of u and a standard error of ohm The mean of the sampling distribution approaches the population so it is the expected value for the sample means Standard Error Standard error of the mean is not the same thing as standard deviation Standard deviation is the standard distance between a score and the mean while standard error is the standard distance between a sample mean and the population mean Standard error measures how much error we should expect between a sample mean and the population mean on average Standard error is often represented as am or SE The SE can be found using one of two equations 0 a o W02n 0 Note This equation is only really useful if you have the variance 02 instead of the standard deviation 0 The larger the sample size the lower the standard error This is because standard error is a function of sample size ZScores for Sample Means You can compute a zscore for the probability of obtaining a certain sample mean These z scores are usually higher than the zscores we found previously because of the extra error the equation includes the standard error Equation for finding zscore for probability of obtaining sample means 2H z CINE Example la In a population with a mean of 80 and a standard deviation of 10 what would the zscore be for a mean score of 90 with nlO Step I 39 List given information u80 x90 610 nlO Step 2 Choose the proper equation iu CINE Step 3 Plug in known quantities Z 90 80 101 0 Step 4 Solve for unknown quantity 10 z 103 16 Subtract 80 from 90 and take square root of 10 Divide 10 by 316 to get 316 in denominator z 316 Divide 10 by 316 to get316 Example 1b What is the probability of getting a sample of z 316 or higher Builds o previous example Step I 39 Draw a graph with the zscore s and mean labeled Note This is a sampling distribution O 316 Step 2 Look up the probability associated with that place on the curve ie zscore using the unit normal table Note We will use the C column because we are looking for the probability of the tail of the distribution as illustrated on the graph above however poorly It is di icult to see the area we are looking for because it is so small A B c D 315 14991 I 4992 3m 31
Are you sure you want to buy this material for
You're already Subscribed!
Looks like you've already subscribed to StudySoup, you won't need to purchase another subscription to get this material. To access this material simply click 'View Full Document'