Here's a list of common calculator instructions for all stats classes.
Prof Kiernanpatty@helpwithnumbers.netcalculatorinstrucitonsforstats3
 Hey everyone,   There will be a workshop on probability in Stats I on Tues. OCT 19, 2010 at 3:20 in the IRC room 132.   Click here to RSVP for a seat and preferred candy type.   -Patty   If the link doesn't work for you here's the address for the form: http://www.zoomerang.com/Survey/WEB22BAPCDV64B
Prof Kiernanpatty@helpwithnumbers.networkshoponprobabilitywordproblems3
 If you're working on chapter 2 you need to know how to construct frequency tables. I've already covered the basics of how to create the tables but here's another example and format of explaining it.I've attached a powerpoint presentation with another step by step example of how to create a frequency table.
Prof Kiernanpatty@helpwithnumbers.netfrequencytablesrevisited3

Weighted means, Standard deviations, and variances

Using just one list:

1. First put values in the X list and get a total

 X 18 18 20 19 21 19 22 21 25 28 24 18 19 20 19 19 21 19 22 32 36 28 total: 488

2. find the mean (average): NOTE: Don’t panic about the notation, there are multiple ways of writing mean.

3. Subtract the mean from every value in the x column:

 X x-mean 18 18 -22.2 = -4.2 18 18 -22.2 = -4.2 20 20 -22.2 = -2.2 19 19 -22.2 = -3.2 21 21 -22.2 = -1.2 19 19 -22.2 = -3.2 22 22 -22.2 = -0.2 21 21 -22.2 = -1.2 25 25 -22.2 = 2.8 28 28 -22.2 = 5.8 24 24 -22.2 = 1.8 18 18 -22.2 = -4.2 19 19 -22.2 = -3.2 20 20 -22.2 = -2.2 19 19 -22.2 = -3.2 19 19 -22.2 = -3.2 21 21 -22.2 = -1.2 19 19 -22.2 = -3.2 22 22 -22.2 = -0.2 32 32 -22.2 = 9.8 36 36 -22.2 = 13.8 28 28 -22.2 = 5.8 total: 488

4. square the x-mean column and get the total of the column:

 X x-mean (x-mean)2 18 18 -22.2 = -4.2 17.64 18 18 -22.2 = -4.2 17.64 20 20 -22.2 = -2.2 4.84 19 19 -22.2 = -3.2 10.24 21 21 -22.2 = -1.2 1.44 19 19 -22.2 = -3.2 10.24 22 22 -22.2 = -0.2 0.04 21 21 -22.2 = -1.2 1.44 25 25 -22.2 = 2.8 7.84 28 28 -22.2 = 5.8 33.64 24 24 -22.2 = 1.8 3.24 18 18 -22.2 = -4.2 17.64 19 19 -22.2 = -3.2 10.24 20 20 -22.2 = -2.2 4.84 19 19 -22.2 = -3.2 10.24 19 19 -22.2 = -3.2 10.24 21 21 -22.2 = -1.2 1.44 19 19 -22.2 = -3.2 10.24 22 22 -22.2 = -0.2 0.04 32 32 -22.2 = 9.8 96.04 36 36 -22.2 = 13.8 190.44 28 28 -22.2 = 5.8 33.64 total: 488 493.28

5.Plug and chuck:

To get the Variance:

AKA: Divide the total of the (x-mean)2 column by the sample size minus 1 Standard Deviation:

AKA: get the square root of what you get from the variance formula.

Prof Kiernanpatty@helpwithnumbers.netuntitledpost-17
So it seems that most people are having an issue with regular probabilities using a table so here's how I'd do a problem with a table of values
I went around campus and asked people who had laptops: if they had a MAC or PC, and if they owned an iPad/iPod/iPhone and got the following values:

 MAC computer PCcomputer Total iPad 5 2 7 iPod 6 4 10 iPhone 3 10 13 Total 14 16 30

All I need to do to find any basic probability is figure out what I'm talking about then divide it by a total. The trick is figuring out what that total is supposed to be.

Any time I have a single word inside the probability i will most likely use the total of the whole group as the denominator (bottom of the fraction). For example:

If I wanted to find the probability of having a MAC computer I'd get the following:
P(MAC) = the number of MAC users/ the overall total = 14/30
If I wanted to find the probability of having a PC computer I'd get the following:
P(PC) = the number of PC users/ the overall total =16/30
If I wanted to find the probability of having an iPad I'd get the following:
If I wanted to find the probability of having an iPod I'd get the following:
P(iPod) = the number of iPod users/ the overall total= 10/30
If I wanted to find the probability of having an iPhone I'd get the following:
P(iPhone) = the number of iPhone users/ the overall total= 13/30

Now If I wanted to find the probability of owning an iPod or a PC I'd use a different set of rules. Then you've got to figure out what this formula means P(A or B)=P(A)+P(B)-P(A and B) ...The formula is basically saying that since I am saying one OR the other I can add the probabilities together and make sure I don't count things twice!

If I wanted to find the probability of owning an iPod or a PC I'd do the following:
P(iPod)= 10/30
P(PC)= 16/30
P(Both iPod and PC)= the only value in both categories / the overall total= 4/30
P(iPod OR PC)= P(iPod) + P(PC) - P(Both) = (10/30)  +  (16/30)  - (4/30) = 22/30
OR I could just count all values in the ipod or the PC categories without counting any numbers twice and divide that by the total to get the probability of owning an iPod or a PC... (2+4+10+6)/30 = 22/30

Prof Kiernanpatty@helpwithnumbers.netprobabilitypart12
 Here is a sample of how you can use a frequency distribution to make graphs.
Prof Kiernanpatty@helpwithnumbers.netfrequencydistributionsandgraphing1
 Nominal Level of Measurement: categories that cannot be put in a universal, meaningful order. Ordinal Level of Measurement: categorical data that can be put in an order, but any numerical differences are meaningless.Interval Level of Measurement: numerical data with no natural starting point Ratio Level of Measurement: numerical data with a natural starting point.

Statistics can be defined as the science of data. The study of statistics is the universal process of data generation, analysis, presentation, and even how to interpret the data. This means statistics is not like any other math class you've ever taken before. The Urbandictionary.com put it simplest when they defined statistics as:

"The math course that is essentially the lovechild mathematics and English... And sometimes psychology."(2011)

Statistics is much more than gambling and surveys. Everyone uses statistics daily basis even without realizing it. Statistics is a course where the numbers are not nearly as important as the thought process used to generate the numbers. Essentially, the best statisticians are skeptical of all computations where raw data is not present and are looking for flaws in data generation preparation, analysis, and conclusions being made with the data. The following topics include common concerns people have when deciding which statistics are flawed.

## Prepare Data

To prepare the data for use you must consider the answers to a series of questions to avoid wasting time analyzing raw data that is flawed. The most important questions used for preparing data are context based questions: What do the data mean? What is the goal of the study?  You should then, consider the source of the data: Is the source objective? Is the source biased? The key idea here is to be vigilant and skeptical of studies from sources that may be biased. Sampling Methods must be considered as well: Does the method chosen greatly influence the validity of the conclusion? Voluntary response (or self-selected) samples often have bias (those with special interest are more likely to participate). Are other methods are more likely to produce good results?

## Analyze Data

The first step of actual data analysis is to create the appropriate graphs (covered in chapter 2). Once these graphs have been created, you should apply statistical methods (the rest of the book explains the statistical methods). Most of the formulas required to compute the numerical values are extremely daunting (some are not possible by hand), thus statisticians rely heavily on technology (computers, graphing calculators, tables). With technology, good analysis does not require strong math skills, but it does require using common sense and paying attention to sound statistical methods.

## Make Conclusions

The whole point of statistical analysis is to decide if the data is significant (different from normal data). Data that is statisticly significant will not happen based on coincidence. Occasionally data can be statistically significant without being practically significant (useful in the real world).

PEOPLE MISUSE STATISTICS
Simply put, people dont understand statistics use them to prove points all of the time. When people don't use the correct methods their conclusions end up being flawed. Below is a list of the most common misuses of Statistics.

MISLEADING CONCLUSIONS (Correlation does not imply causation)

Concluding that one variable causes the other variable when in fact the variables are only correlated or associated together (covered in Chapter 10). In essence, two variables that may seem to be related, are temperature and violent crimes (as it gets hotter outside, the number of violent crimes will increase). However, we cannot conclude the one causes the other based solely on the numerical calculation of the relationship between temperature and the number of violent crimes. There may be another factor involved (like discomfort) that explains the relationship. Which is where the mantra used in many social science classes "Correlation does not imply causation" comes from.

SMALL SAMPLE SIZES
Conclusions should never be based on tiny sample sizes. If you are looking to decide if a study technique works for all college students, and you only study 8 college students, your data is pretty much useless. If your data is to be useful it should be based on a reasonable percentage of the population. In other words, the smaller the sample size (i.e. number of participants in a study) the less useful the data will be. In the real world, you want to have an experiment that has a reasonable number of participants.

For example, if Prof K. surveys 20 people to figure out which Walt Disney World restaurants people like eating at. and comes to the decision that the Be Our Guest restaurant is the most popular counter service restaurant in the Magic Kingdom. Thanks to Google.com Prof. K found that there are approximately 17 MILLION visitors to Walt Disney World every year. What does that say about the validity of Prof. K's data?

If survey questions are not worded well, the results can be misleading.
In a famous Psychological Exp
eriment Elizabeth Loftus & John Palmer found that by changing one word in a question could change the responses of people Click here for more info on this experiment In the experiment participants had to watch several videos of car accidents then each participant was asked "How fast the cars were going when they ____?" Several different words were used to complete the question and the responses changed based on the word used. When participants were asked "How fast were the cars going when they contacted?" the average response was 32 mph, yet when the participants were asked How fast were the cars going when they smashed?" the average response was 42 mph. That's a 10 mph difference based on the same video footage.

Image Source: http://www.simplypsychology.org/loftus-results.jpg

ORDER OF QUESTIONS
The order of the questions can change the results as well. Sometimes, questions are unintentionally loaded by such factors as the order of the items being considered.
For Example: Would you say traffic contributes more or less to air pollution than industry?
Results in: traffic - 45%; industry - 27%
When the order is reversed the results change to: industry - 57%; traffic - 24%

NONRESPONSES
Your data may be flawed due to not having the right mix of people answering the question. This occurs when someone either refuses to respond to a survey question or is unavailable. People who refuse to talk to pollsters have a view of the world around them that is markedly different than those who will let pollsters into their homes, or have the time to answer the polls. Think about it, who has time to answer a poll that takes 15 minutes (without getting reimbursed for your time)?

MISSING DATA
Missing data can change your numbers drastically. People frequently drop out of studies for many reasons that aren't related to the study. Think about how many reasons you could have for missing a final exam in a Biology course. Now think about how your course average would change due to a zero for the final exam. Finally think about what would happen to your average if you missed the Biology midterm and final exams. Simply put, when people drop out of studies they change the statistics.

PRESCISE NUMBERS
Simply put, just because a number is exact, doesn't mean it hasn't been estimated. A number can be an estimate but should always be referred to as an estimate. Think about the many ways people can round any 4 digit number (i.e. 1,675.43 could be rounded to 2,000 or 1,700 or 1,680 these numbers are all correct yet they are all different estimates) Now, think about how you could be manipulated into buying one computer over another computer by someone saying one computer costs \$1,700 vs \$1680.

PERCENTAGES
Simply put, many people don't understand fractions or percentages and misuse them regularly.

Misleading or unclear percentages are sometimes used. Textbook Example – Continental Airlines ran an ad claiming “We’ve already improved 100% in the last six months” with respect to lost baggage. Does this mean Continental made no mistakes?

Prof Kiernanpatty@helpwithnumbers.netdefiningstatisticsstatisticalthinking1
 How to make a Box plot by hand: 1. Find the 5-number summary (smallest value, Q1, Median, Q3, & the largest value) 2. Construct a scale with values that include the minimum and maximum values 3. Construct a box extending from the first quartile to the third quartile and draw a line in the box at the median value 4. Draw lines extending outward from the box to the minimum and the maximum data values and you're done! How to make a Box Plot using the calculator:Put data into list #1: Press STAT Press ENTER (1:Edit) Put in variable under list 1 L1  Press ENTER (continue the last two steps until you have added all of the variables in your list)Get the 5 number summary: Press STAT Press CALC            Press 1 (1: 1-Var Stat)              Press 2nd             Press L1 (or the name of the list that you have put your information in)Press ENTERPress Down arrow (the last 5 things in the list are the 5 number summary)To see the Box Plot:Press 2nd Press Y=Press ENTERPress ENTERPress DOWN ARROW Press RIGHT ARROW (till you highlight the box plot with the dots)Press ENTERPress ZOOMPRESS 9 (9: zoom stat)Voila! You have your box plot!To find the Q1, Median, and Q3 by hand:Put your data in a list from small to largeFind the median (the value in the middle of the dataset) if there are 2 data values average them together to get the medianFind the first and second Quartiles (Q1 & Q3) the same way:make 2 separate data sets and  find the middle value of each new datasets.Find the Inter Quartile Range: Q3-Q1Apply the Interquartile range to the formula below to ensure there are no outliersIf there are outliers use the IQR (inter quartile range) to get the min and max values accepted as normal values Put any outliers on the plot as dots.
Prof Kiernanpatty@helpwithnumbers.netboxplots4

Important terminology:

 Frequency distribution A table of data values and corresponding frequencies. Frequency The number of data points that fall within a class, or range of data points. Class Levels or categories used to explain the spread of a data list Lower class limit (LCL) Smallest whole number boundary of a class Upper class limit (UCL) Largest whole number boundary of a class Class boundary The space between the gap of each class limit Class midpoints The median of each class (add the lower class limit and the upper class limit and divide by 2) Class width The distance between each consecutive lower class limit (the highest value minus the lowest value divided by the number of classes)

How to make a frequency distribution:

1.    Decide how many classes you want to have

2.    Find the class width and round the number (most of the time you round this number up)

3.    Find the lowest data value in the data set, this will be the lowest class limit

4.    Add the class with to the lowest class limit (from step 3). This is the second class’ lower limit.

5.    Repeat step 4 until you exceed the highest value in the data set.

6.    Find the upper class limits, add one minus the class width to each lower limit

7.    Fill out the class section of the frequency distribution (the table below).

8.    Find the class frequencies (number of values that fall within each class).

Relative frequency distributions:

9.    Divide each class frequency by the sum of all frequencies.

Cumulative frequency distributions:

9.  Using the upper class limits, at each limit add all frequencies below the upper class limit

EXAMPLE 1:

I recently asked several tutors what their GPAs were, and got the following GPAs: 3.4, 3.7, 3.0, 3.9, 3.4, 3.1, 3.5, 3.8, 3.0, 3.5, 3.4, 3.4, 3.9, 3.8, 3.9, 3.2, 3.2

If I want to show the info in a frequency distribution. I’d follow these steps:

1.    Decide how many classes you want to have

I want to have 5 classes

2.    Find the class width and round the number (most of the time you round this number up)

Class width: (3.9-3.0) / 5 = 0.18 ≈ 0.2

3.    Find the lowest data value in the data set, this will be the lowest class limit

3.0 is the lowest class limit

4.    Add the class width to the lowest class limit (from step 3). This is the second class’ lower limit.

3.0 + 0.2 = 3.2

5.    Repeat step 4 until you exceed the highest value in the data set.

 Lower Class Limit 3.0 3.0 +0.2 = 3.2 3.2 +0.2 = 3.4 3.4 +0.2 = 3.6 3.6 +0.2 = 3.8 3.8 +0.2 = 4.0

6.    Find the upper class limits, typically one minus the class width to each lower limit.

 Lower Class Limit Upper Class Limit 3.0 3.2 - 0.1 = 3.1 3.0 +0.2 = 3.2 3.4 - 0.1 = 3.3 3.2 +0.2 = 3.4 3.6 - 0.1 = 3.5 3.4 +0.2 = 3.6 3.8 - 0.1 = 3.7 3.6 +0.2 = 3.8 4.0 - 0.1 = 3.9

7.    Fill out the class section of the frequency distribution (the table below).

 Classes Lower Class Limit Upper Class Limit 3 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9

8.    Find the class frequencies (number of values that fall within each class).

 Classes Counts Frequency Lower Class Limit Upper Class Limit 3 3.1 ||| 3 3.2 3.3 || 2 3.4 3.5 |||||| 6 3.6 3.7 | 1 3.8 3.9 ||||| 5

Relative frequency distributions, follow steps 1-8 above and:

9.    Divide each class frequency by the sum of all frequencies.

Relative frequency distribution:

 Classes Frequency Relative Frequency Lower Class Limit Upper Class Limit 3 3.1 3 3/17 = 0.18 3.2 3.3 2 2/17 = 0.12 3.4 3.5 6 6/17 = 0.35 3.6 3.7 1 1/17 = 0.06 3.8 3.9 5 5/17 = 0.29 TOTAL 17 1.00

Cumulative frequency distributions, follow steps 1-8 above and:

9. Using the upper class limits, at each limit add all frequencies below the

Upper Class Limit (see below)

 Classes Frequency Class Cumulative frequency Lower Class Limit Upper Class Limit Less than 3.0 0 3 3.1 3 Less than 3.2 3 3.2 3.3 2 Less than 3.4 3+2= 5 3.4 3.5 6 Less than 3.6 3+2+6= 11 3.6 3.7 1 Less than 3.8 3+2+6+1= 12 3.8 3.9 5 Less than 4.0 3+2+6+1+5= 17

Prof Kiernanpatty@helpwithnumbers.netuntitledpost3
 I have attached a flow chart that should help you figure out which level of measurement fits your data. Remember you are not concentrating on the data counts, but the labels associated with the data. For example:9.8 inches of snow7 large bags of popcorn8 Middlesex County College Students27º Fahrenheit For these 4 samples you should ignore the numbers, you are concerned with the labels, not the numbers. inches of snowlarge bags of popcornMiddlesex County College StudentsFahrenheit Answers:1. Ratio, 2.Ordinal, 3. Nominal, 4. Interval
Prof Kiernanpatty@helpwithnumbers.netlevelsofmeasurementflowchart3
https://sites.google.com/feeds/content/helpwithnumbers.net/www/12029123621294294322010-10-19T01:09:46.861Z2016-09-03T02:27:10.826Z2014-02-06T14:04:40.648ZEmpiracle rule & Chebyshev's rule translated in English
Prof Kiernanpatty@helpwithnumbers.netempiraclerulechebyshevsruletranslatedinenglish4
https://sites.google.com/feeds/content/helpwithnumbers.net/www/37215858771308717732014-03-18T01:22:52.602Z2016-09-03T02:27:10.418Z2014-03-18T01:22:52.597ZFinding the mean and variance of a probability distribution
To find the variance of a probability distribution you need to use this formula:
In English this means you need to do the following:
First: subtract the mean from each x value and square each answer
Second: Multiply each answer from the first step by each probability
Third: get the sum of the answers from the second step

Note: remember the standard deviation is the square root of the variance.

Example 1: Find the mean and variance of the following probability distribution.

 x P(x) 0 0.1 1 0.3 2 0.4 3 0.2

To find the mean of the distribution we need to add another vertical column onto our table and a total row at the bottom of our table. and compute the x*P(x) for each x value then get the total of the column as our mean.

 x P(x) x*P(x) 0 0.1 0*0.1=0 1 0.3 1*0.3=0.3 2 0.4 2*0.4=0.8 3 0.2 3*0.2=0.6 Total 1.7

Thus the mean of example 1 is  1.7 .

To find the variance of the distribution we need to add 3 new vertical columns onto our original table and a total row at the bottom of the table. The computations in each of the new columns are as follows:
In the first new column, subtract the mean from each x value (remember our mean is  1.7 )
 x P(x) x-mean 0 0.1 0 -  1.7  = -1.7 1 0.3 1 -  1.7  = -0.7 2 0.4 2 -  1.7  = 0.3 3 0.2 3 -  1.7  = 1.3 Total

In the second new column, square each answer from the first new column.
 x P(x) x-mean (x-mean)2 0 0.1 0 -  1.7  = -1.7 (-1.7)2 = 2.89 1 0.3 1 -  1.7  = -0.7 (-0.7)2 =0.49 2 0.4 2 -  1.7  = 0.3 0.32 =0.09 3 0.2 3 -  1.7  = 1.3 1.32 =1.69 Total

In the third new column, multiply each answer from the second new column by each probability and finally get the sum of the answers from this step.
 x P(x) x-mean (x-mean)2 (x-mean)2*P(x) 0 0.1 0 -  1.7  = -1.7 (-1.7)2 = 2.89 2.89*0.1 = 0.289 1 0.3 1 -  1.7  = -0.7 (-0.7)2 =0.49 0.49*0.3 =0.147 2 0.4 2 -  1.7  = 0.3 0.32 =0.09 0.09*0.4 =0.036 3 0.2 3 -  1.7  = 1.3 1.32 =1.69 1.69*0.2 = 0.338 Total 0.81

Thus our variance for example 1 is 0.81. Which means our standard deviation for this example is the square root of our variance or:

Prof Kiernanpatty@helpwithnumbers.netfindingthemeanandvarianceofaprobabilitydistribution1
 As always, DON'T let the technical jargon confuse you. Prof Kiernanpatty@helpwithnumbers.netsummationnotation3
https://sites.google.com/feeds/content/helpwithnumbers.net/www/67264595456285200112014-05-04T03:57:57.111Z2016-09-03T02:27:09.691Z2014-05-04T03:57:57.106ZImportant things to remember when you get to Stats 2
 Here are some VITAL concepts you should remember when you get to Stats 2.This information may be helpful while you are preparing for the final.
Prof Kiernanpatty@helpwithnumbers.netimportantthingstorememberwhenyougettostats21 Key words: Conditional probability, given, P(A|B), using tables to find probabilities
Prof Kiernanpatty@helpwithnumbers.netprobabilitypart2conditionalprobability3
 Here's a copy of the suggested formula sheet for Exam 3. Make sure you know how to use the formulas on this sheet for the exam.
Prof Kiernanpatty@helpwithnumbers.netexam3formulasheet1
 Here's a PDF and PowerPoint version of the information covered on the first day or two of stats 1.   According to the Math department you must know at least the following information: Students should understand the difference between sample, population, statistic and parameter. Students should be able to distinguish between quantitative data and qualitative data. Students should be able to distinguish between discrete data and continuous data. Students should be able to identify the levels of measurement: nominal, ordinal, ratio and interval. Students should be able to recognize the different methods of sampling: random, stratified, cluster, systematic, and convenience. I have covered all of the above topics in the attachments below.
Prof Kiernanpatty@helpwithnumbers.netintrotostatistics12
https://sites.google.com/feeds/content/helpwithnumbers.net/www/79213951480761366032014-01-31T21:51:53.117Z2016-09-03T02:27:07.139Z2014-02-06T01:58:25.017ZFinding Weighted Means and Averages For Frequency Distributions
Find the mean of this frequency distribution:

 class frequency 10-19 8 20-29 16 30-39 21 40-49 11 50-59 4

1. The first thing you need to do is to find the midpoint for each class.
 Class Midpoint 10-19 (10+19)/2 = 14.5 20-29 (20+29)/2 = 24.5 30-39 (30+39)/2 = 34.5 40-49 (40+49)/2 = 44.5 50-59 (50+59)/2 = 54.5

2. Next multiply each midpoint by the corresponding frequency and get the total.
 Midpoint Frequency Midpoint * frequency 14.5 8 14.5 * 8 =    116 24.5 16 24.5 * 16 =    392 34.5 21 34.5 * 21 = 724.5 44.5 11 44.5 * 11 = 489.5 54.5 4 54.5 * 4 =    218 Total: 60 Total: 1940

3. Finally divide the total of the Midpoint * frequency column by the total of the frequency column to get your mean.
1940 / 60 = 32.333333333...
Always round to one value beyond what your original data was (since our classes were whole numbers we should round to 1 decimal place). So our final answer for the mean of this frequency distribution is 32.3.
Prof Kiernanpatty@helpwithnumbers.netfindingweightedmeansandaveragesforfrequencydistributions3
https://sites.google.com/feeds/content/helpwithnumbers.net/www/24089937918299021662014-03-17T23:50:41.324Z2016-09-03T02:27:07.100Z2014-03-17T23:50:41.290ZChecking a probability distribution for validity
When you are asked if a probability distribution (table) is valid you need to answer 3 questions.

1. Does the sum of P(x) add up to any number other than 1 ?
2. Are there any negative probabilities?
3. Are there any probabilities larger than 1?

If you answer YES to any of the questions above your table is NOT a probability distribution.
If you answer NO to all of the questions above your table is a probability distribution.

Example 1:
 x P(x) 0 0.129 1 0.257 2 0.659 3 0.008 4 -0.053

1. No, the sum of P(x) adds up to 1
0.129+0.257+0.659+0.008+(-0.053) = 1.000
2. Yes, there are negative probabilities.
3. No, there not probabilities larger than 1.
Since we said there are negative probabilities, example 1 is NOT a probability distribution.

Example 2:
 x P(x) -2 0.2 -1 0.2 0 0.2 1 0.2 2 0.2

1. No, the sum of P(x) adds up to 1
0.2+0.2+0.2+0.2+0.2 = 1.000
2. No, there aren't any negative probabilities. The negative numbers are values for x, not the probability of x.
3. No, there aren't any probabilities larger than 1.
Since we said No to all of the questions, example 2 is a probability distribution.

Example 3:
 x P(x) 1 0.200 2 0.200 3 0.200 4 0.200 5 0.199

1. Yes, the sum of P(x) adds up to 0.999
0.2+0.2+0.2+0.2+0.199 = 0.999
2. No, there aren't any negative probabilities. The negative numbers are values for x, not the probability of x.
3. No, there aren't any probabilities larger than 1.
Since we said Yes to the first question, example 3 is NOT a probability distribution.

Prof Kiernanpatty@helpwithnumbers.netcheckingaprobabilitydistributionforvalidity1
Prof Kiernanpatty@helpwithnumbers.netterminologyofmeasurementvariables2
https://sites.google.com/feeds/content/helpwithnumbers.net/www/41742004552294357802014-01-26T21:53:54.796Z2016-09-03T02:27:04.439Z2014-01-26T21:53:54.792ZCalculator Shortcuts for graphs and histograms
 To Enter data into a list: Press stat Press enter Put data into a list (remember the list number that displays at the top of the screen L1, L2, L3, L4, L5, or L6) To view graphsPress 2nd Press Y= (its just below your screen) Press enter Press enter Press the down arrow    Use your left and right arrow keys to highlight the histogram or type of graph you want to see Once you’ve highlighted the correct graph press enter Press the down arrow  the Xlist: should say the list number (to type in a list number you need to press 2nd  then the number of the list Freq should be: 1   Press zoom Press 9 Press trace Use your left and right arrow keys to view the different values for the histogram or graph Min = is the lower limit Max < is the upper limit n = is the frequency for that class The 2 screens  on your calculator will look like this:To clear a list:   Press stat Press 4Press 2nd Press the number of the list you want to clearPress enter
Prof Kiernanpatty@helpwithnumbers.netcalculatorshortcutsforgraphsandhistograms1

Steps for Hypothesis testing:

1.     Write down what’s given

i.e. sample standard deviation, sample size, population proportion

2.     Figure out what table & formula you should use

3.     Draw the picture!

Left tail is less than

Right tail is greater than

Two tail is equal to or not equal to

4.     Use the tables to find the critical value and add it to the picture

Use α to find the critical value in a one tailed test

Use, 𝛼/2 to find the critical value in a two tailed test

5.     Write the hypothesis

a.     The null hypothesis H0 always has an equal sign

b.     The alternative hypothesis has either a less than sign <, greater than sign> or not equal sign

6.     Use formula to find the test statistic Z, t, X 2 etc.

7.     Decide whether to reject or fail to reject the null hypothesis.

If the test statistic falls in the shaded critical region, reject the null hypothesis

If the test statistic does NOT fall in the shaded critical region, “Fail to reject” the null hypothesis

 Original claim is the null hypothesis Original claim is alternative hypothesis Reject H0 “There is sufficient evidence to warrant rejection of the claim that… (original claim)” This is the only time that the original claim can be rejected. “The sample data supports the claim that…(original claim)” This is the only time that the original claim can be supported Fail to reject H0 “There is NOT sufficient evidence to warrant rejection of the claim that… (original claim)” “There is NOT sufficient sample evidence to support the claim that…(original claim)”

Source:  Triola, M. F. (2003). Elementary Statistics (9th ed.), Pearson Education, Inc.

Prof Kiernanpatty@helpwithnumbers.nethypothesistesting1013
https://sites.google.com/feeds/content/helpwithnumbers.net/www/73468584865509654072010-03-04T18:30:07.424Z2016-09-03T02:27:04.270Z2010-03-23T23:33:02.541ZIntro to Probability, Sample spaces, and probabilty distributions
·         A sample space is a “set of all possible outcomes or events in an experiment that cannot be further broken down” (Triola, 2008).

·

A probability distribution is “a collection of values of a random variable along with their corresponding probabilities” (Triola, 2008).

• There are no negative probabilities.
• There are no probabilities larger than one.
• The total of all probabilities must add up to one.

• A sample space has no probabilities.

Factor tree method

Example 1: If we wanted to find the probability of flipping a coin 3 times and having 0,1,2 or 3 heads we could use a factor tree to show the possible combinations.

To create a factor tree, we must start with the first possibility (heads or tails in blue). Then the second coin possibilities (in red) branch off of each of the first coin toss. Finally, the third coin (in green) branches off of each of the second coin toss.

Sample space:

Probability distribution:

Example 2:

If you were to have 2 children, what would the probability of having zero, one, or two boys be?

To answer this question the first thing we need is to define the sample space. That is, we need to create a list of all possible combinations of children. Illustrate that you can have a boy then another boy, or a boy then a girl, or a girl then a boy, or a girl then another girl.

S = {Boy Boy, Boy Girl, Girl Boy, Girl Girl}

Next write the probability distribution on the board showing where each of the probabilities come from (combinations column):

 Combinations Number of boys in 2 births (x) P (x) Girl Girl 0 ¼ Boy Girl Girl Boy 1 ½ Boy Boy 2 ¼

Prof Kiernanpatty@helpwithnumbers.netintrotoprobabilitysamplespacesandprobabiltydistributions2
https://sites.google.com/feeds/content/helpwithnumbers.net/www/43370248662285303842010-12-02T21:37:45.801Z2016-09-03T02:27:04.056Z2011-02-02T01:52:02.426ZUseful tables for probability of cards and dice
Here is a useful table for finding probability of a deck of cards:

 Hearts Ace 2 3 4 5 6 7 8 9 10 Jack Queen King Diamonds Ace 2 3 4 5 6 7 8 9 10 Jack Queen King Clubs Ace 2 3 4 5 6 7 8 9 10 Jack Queen King Spades Ace 2 3 4 5 6 7 8 9 10 Jack Queen King

Here is a useful table for finding probability of rolling 2 dice:

 1 2 3 4 5 6 1 1,1 2,1 3,1 4,1 5,1 6,1 2 1,2 2,2 3,2 4,2 5,2 6,2 3 1,3 2,3 3,3 4,3 5,3 6,3 4 1,4 2,4 3,4 4,4 5,4 6,4 5 1,5 2,5 3,5 4,5 5,5 6,5 6 1,6 2,6 3,6 4,6 5,6 6,6

Prof Kiernanpatty@helpwithnumbers.netusefultablesforprobabilityofcardsanddice3
 An alternate explanation of why having a strong correlation doesn't imply causation. Source: XKCD Comics