Modeling the relationship between a scalar variable y and one or more variables denoted X. In linear regression, models of the unknown parameters are estimated from the data using linear functions.
polyfit( x,y2,1) %return 2.1667 -1.3333, i.e 2.1667x-1.3333
The null hypothesis (denote by H0 ) is a statement about the value of
a population parameter (such as mean), and it must contain the condition of equality and must be written with the symbol =, ≤, or ≤.
As the sample size increases, the sampling distribution of sample
means approaches a normal distribution
If all possible random samples of size n are selected from a population with mean μ and standard deviation σ, the mean of the sample means is denoted by μ x̄ , so
μ x̄ = μ
the standard deviation of the sample means is:
σ x̄ = σ⁄√ n
A hash table is a data structure used to implement an associative array, a structure that can map keys to values. A hash table uses a hash function to compute an index into an array of buckets or slots, from which the correct value can be found.
For binary search, the array should be arranged in ascending or descending order. In each step, the algorithm compares the search key value with the key value of the middle element of the array. If the keys match, then a matching element has been found and its index, or position, is returned. Otherwise, if the search key is less than the middle element's key, then the algorithm repeats its action on the sub-array to the left of the middle element or, if the search key is greater, on the sub-array to the right.
P(x)= p x q n-x n!/[(n-x)!x!]
where n = number of trials
x = number of successes among n trials
p = probability of success in any one trial
q = 1 -p
Given that the population of men has normally distributed weights, with a mean of 173 lb and a standard deviation of 30 lb, find the probability that
a. if 1 man is randomly selected, his weight is greater than 180 lb.
b. if 36 different men are randomly selected, their mean weight is greater that 180 lb.
Solution: a) z = (x - μ)/ σ = (180-173)/30 = 0.23
For normal distribution P(Z>0.23) = 0.4090
b) σ x̄ = σ/√n = 20/√ 36 = 5
z= (180-173)/5 = 1.40
P(Z>1.4) = 0.0808
The probability of rejecting the null hypothesis when it is called
the significance level α , and very common choices are
α = 0.05 and α = 0.01
The Alternative hypothesis (denoted by H1 ) is the statement that must be true if the null hypothesis is false.
T-test is any statistical hypothesis test in which the test statistic follows a Student's t distribution if the null hypothesis is supported.
[h,p,ci] = ttest(y2,0)% return 1 0.0018 ci =2.6280 7.0863
Measure of how much two variables change together
y2=[1 3 4 5 6 7 8]
cov(x,y2) %return 2*2 matrix, diagonal represents variance
Quantitative measure of the shape of a set of points.
moment(x, 2); %return second moment
Kurtosis is a measure of how outlier-prone a distribution is.
kurtosis(x) % return2.3594
Describes how far values lie from the mean
var(x) %return 1.1429
Skewness is a measure of the asymmetry of the data around the sample mean. If skewness is negative, the data are spread out more to the left of the mean than to the right. If skewness is positive, the data are spread out more to the right.
Skewness(x) % return-0.5954
► first quartile (25th percentile)
► second quartile (50th percentile)
► third quartile (75th percentile)
► kth percentile
► prctile(x, 25) % 25th percentile, return 2.25
► prctile(x, 50) % 50th percentile, return 3, i.e. median
Median is described as the numeric value separating the higher half of a sample, a population, or a probability distribution, from the lower half. The median of a finite list of numbers can be found by arranging all the observations from lowest value to highest value and picking the middle one
median(x) % return 3.
The mode of a data sample is the element that occurs most often in the collection.
x=[1 2 3 3 3 4 4]
mode(x) % return 3, happen most
There are four sampling methods:
► Simple Random (purely random),
► Systematic( every kth member of population),
► Cluster (population divided into groups or clusters)
► Stratified (divided by exclusive groups or strata, sample from each group) samplings.
Sampling is that part of statistical practice concerned with the selection of an unbiased or random subset of individual observations within a population of individuals intended to yield some knowledge about the population of concern.
Suppose that the experimental results show the coin turning up heads 14 times out of 20 total flips
► null hypothesis (H0): fair coin;
► observation O: 14 heads out of 20 flips; and
► p-value of observation O given H0 = Prob(≥ 14 heads or ≥ 14 tails) = 0.115.
The calculated p-value exceeds 0.05, so the observation is consistent with the null hypothesis - that the observed result of 14 heads out of 20 flips can be ascribed to chance alone - as it falls within the range of what would happen 95% of the time were this in fact the case. In our example, we fail to reject the null hypothesis at the 5% level. Although the coin did not fall evenly, the deviation from expected outcome is small enough to be reported as being "not statistically significant at the 5% level".
In statistical significance testing, the p-value is the probability of obtaining a test statistic at least as extreme as the one that was actually observed, assuming that the null hypothesis is true. If the p-value is less than 0.05 or 0.01, corresponding respectively to a 5% or 1% chance of rejecting the null hypothesis when it is true
The probability of some observed outcomes given a set of parameter values is regarded as the likelihood of the set of parameter values given the observed outcomes.
Frequentists condition on a hypothesis of choice and consider the probability distribution on the data, whether observed or not.
Bayesians condition on the data actually observed and consider the probability distribution on the hypotheses.
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