Celeste Sanoria
Summary
Math 6
Lesson 1. Random variables, probability distributions
The probability distribution for a random variable describes how the probabilities are distributed over the values of the random variable. For a discrete random variable, x, the probability distribution is defined by a probability mass function, denoted by f(x)
This function provides the probability for each value of the random variable. In the development of the probability function for a discrete random variable, two conditions must be satisfied: (1) f(x) must be nonnegative for each value of the random variable, and (2) the sum of the probabilities for each value of the random variable must equal one.
Lesson 2: Measures of central tendency and variability; higher-order moments
Central tendency- the tendency of quantitative data to cluster around some central value. The closeness with which the values surround the central value is commonly quantified using the standard deviation.
Variability (dispersion) - is contrasted with location or central tendency, and together they are the most used properties of distributions. They allow us to summarize our data set with a single value hence giving a more accurate picture of our data set.
Lesson 3:
In order to analyze the correlation between two variables, we consult bivariate statistics to search out the embedded information.
Pearson sample correlation coefficient measures the strength of any linear relationship between two numerical variables.
Formula:
X+Y / square root of (x2 + y2)
Example:
X is the # of people in room 1
Y is the # of people in room 2
Room 1 (x) has 5 people
Room 2 (y) has 7 people
To find the correlation in this 2 rooms in the number of people each at the same time , we will use the formula above.
X+Y divide by the product of the square root of x2 + y2
5+7 / square roof of (25+49)
= 12 / SQAURE ROOT OF 74
= 12 / 8.60
= 1.40
Its value can range from -1 for a perfect negative linear relationship to +1 for a perfect positive linear relationship. A value of 0 (zero) indicates no relationship between the two variables.
Lesson 4
Joint distribution
the concept of conditional probability - The term “conditional” refers to the fact that we will have additional conditions, restrictions, or other information when we are asked to calculate this type of probability.
Example:
Number of students who passed or failed in section Blessed
Passed Failed Total
30 15 45
Probability of the students who passed over 45 students in one section is
P = 30/45
P = .67 or 67%
Percentage of failed students :
P = 15/45
P = .33
LESSON 5
Hypothesis testing refers to the process of choosing between two hypothesis statements about a probability distribution based on observed data from the distribution.
The methodology behind hypothesis testing:
LESSON 6
State the null hypothesis
In this step, you set up two statements to determine the validity of a statistical claim: a null hypothesis and an alternative hypothesis.
The null hypothesis is a statement containing a null, or zero, difference. It is the null hypothesis that undergoes the testing procedure, whether it is the original claim or not. The notation for the null hypothesis H0 represents the status quo or what is assumed to be true. It always contains the equal sign.
The alternative statement must be true if the null hypothesis is false. An alternative hypothesis is represented as H1. It Is the opposite of the null and is what you wish to support. It also never contains the equal sign.
Lesson 7
Type I and II errors
Type I - This error occurs when we reject the null hypothesis when we should have retained it. That means that we believe we found a genuine effect when in reality there isn’t one.
Threshold of probability is 0.05 or 5%
Type II - This error occurs when we fail to reject the null hypothesis. In other words, we believe that there isn’t a genuine effect when actually there is one. The probability of a Type II error is represented as β and this is related to the power of the test (power = 1- β). Cohen (1998) proposed that the maximum accepted probability of a Type II error should be 20% (β = 0.2).
Lesson 8
Determine the rejection and non-rejection regions
In this step we calculate the significance level . The significance level, also denoted as alpha or α, is the probability of rejecting the null hypothesis when it is true. For example, a significance level of 0.05 indicates a 5% risk of concluding that a difference exists when there is no actual difference.
Determine the value of the test statistics
The values of the test statistic separate the rejection and non-rejection regions.
Rejection region: the set of values for the test statistic that leads to rejection of H0.
Non-rejection region: the set of values not in the rejection region that leads to non-rejection of H0.
Lesson 9
The P-Value: Another quantitative measure for reporting the result of a test of hypothesis is the p-value. It is also called the probability of chance in order to test. The lower the p-value the greater likelihood of obtaining the same result. And as a result, a low p-value is a good indication that the results are not due to random chance alone. P-value = the probability of obtaining a test statistic equal to or more extreme value than the observed value of H0. As a result H0 will be true.
We then compare the p-value with α:
Lesson 10 : Tests for the expected value and variance of random variables
Rules for normally distributed data
where μ is the expected value of the random variables, σ equals their distribution's standard deviation divided by n1/2, and n is the number of random variables.
How do you calculate standard deviation from standard error?
First, take the square of the difference between each data point and the sample mean, finding the sum of those values. Then, divide that sum by the sample size minus one, which is the variance. Finally, take the square root of the variance to get the SD.
Example Problem
You grow 20 crystals from a solution and measure the length of each crystal in millimeters. Here is your data: 9, 2, 5, 4, 12, 7, 8, 11, 9, 3, 7, 4, 12, 5, 4, 10, 9, 6, 9, 4
Calculate the sample standard deviation of the length of the crystals.
Calculate the mean of the data. Add up all the numbers and divide by the total number of data points.(9+2+5+4+12+7+8+11+9+3+7+4+12+5+4+10+9+6+9+4) / 20 = 140/20 = 7
Subtract the mean from each data point (or the other way around, if you prefer... you will be squaring this number, so it does not matter if it is positive or negative).(9 - 7)2 = (2)2 = 4
(2 - 7)2 = (-5)2 = 25
(5 - 7)2 = (-2)2 = 4
(4 - 7)2 = (-3)2 = 9
(12 - 7)2 = (5)2 = 25
(7 - 7)2 = (0)2 = 0
(8 - 7)2 = (1)2 = 1
(11 - 7)2 = (4)22 = 16
(9 - 7)2 = (2)2 = 4
(3 - 7)2 = (-4)22 = 16
(7 - 7)2 = (0)2 = 0
(4 - 7)2 = (-3)2 = 9
(12 - 7)2 = (5)2 = 25
(5 - 7)2 = (-2)2 = 4
(4 - 7)2 = (-3)2 = 9
(10 - 7)2 = (3)2 = 9
(9 - 7)2 = (2)2 = 4
(6 - 7)2 = (-1)2 = 1
(9 - 7)2 = (2)2 = 4
(4 - 7)2 = (-3)22 = 9
Calculate the mean of the squared differences.(4+25+4+9+25+0+1+16+4+16+0+9+25+4+9+9+4+1+4+9) / 19 = 178/19 = 9.368
This value is the sample variance. The sample variance is 9.368
The population standard deviation is the square root of the variance. Use a calculator to obtain this number.(9.368)1/2 = 3.061The population standard deviation is 3.061
Lesson 11
Hypothesis testing relates to a single conclusion of statistical significance vs. no statistical significance.
Confidence intervals provide a range of plausible values for your population.
Confidence intervals gives us a range of possible values and an estimate of the precision for our parameter value. Hypothesis tests tells us how confident we are in drawing conclusions about the population parameter from our sample.
Lesson 12
What's the difference between statistical and practical significance? While statistical significance shows that an effect exists in a study, practical significance shows that the effect is large enough to be meaningful in the real world.
