Examples of discrete raw data :

Number of siblings of students in S18D
1,2,4,5,8,1,3,4,7,4,2,5,4,6,3,4

• Number of cars passing a checkpoint in 30 minutes
• The shoe sizes of children in a class
  
  

Why raw?

• They have not been ordered in any way
  
  

Characteristic of discrete data

• Can take only exact values
  
  

Bar Chart

• bar - represents frequency

Mode - value that occurs most often, most popular,a variable with highest frequency

Examples of continuous raw data :

• The speed of a vehicle as it passes a checkpoint
• The mass of a cooking apple
• The time taken by a volunteer to perform a task.
  
  

Characteristic of continuous data

• cannot take exact values but can give only within specified range/ measured to a specified degree of accuracy

Way 1 to write the
same set of intervals :
Height (cm)
119.5\ \le h\ <\ 124.5\
124.5 < h < 129.5
129.5 < h < 134.5
134.5 < h < 139.5
139.5 < h < 144.5

Way 3 to write the
same set of intervals :
Height (to the nearest cm)
120 -124
125 - 129
130 - 134
135 - 139
140 - 144

The upper class boundary of one interval is the lower class boundary of the next interval

Width of an interval = Upper class boundary - Lower class boundary

$$A\,\cup\,B\,(A\,or\,B\,or\,both)$$
$$P(A\,\cup\,B)=P(A)+P(B)-P(A\,\cap\,B)$$

For independent events, P(A|B)=P(A)

For independent events, \[P(A\,\cap\,B)=P(A)\,\times\,P(B)\]

Combination
\[^{n}C_{r}\,or\,_{n}C_{r}=\frac{n!}{n!(n-r)!}\]

Unbiased estimator of \({\sigma^2}\)
\[S_{n-1}^2=\frac{n}{n-1}S_{n}\]

• An estimator is unbiased if its expected value equals the population mean of X
• The most efficient estimator has the smallest variance

Continuous random variable with probability density function f(x)\[E(g(X))=\int g(x)f(x)dx\]\[E(X)=\int xf(x)dx\]

For Sample mean
\[E(\bar{X})=E(X)\]\[Var(\bar{X})=\frac{Var(X)}{n}\]

For sample sum\[E(\sum_{i=1}^{n}X_{i})=nE(X)\]\[Var(\sum_{i=1}^{n}X_{i})=nVar(X)\]

\[E(a_{1}X_{1}\pm a_{2}X_{2}\pm a_{3})=a_{1}E(X_{1})\pm a_{2}E(X_{2})\pm a_{3} \]\[Var(a_{1}X_{1}\pm a_{2}X_{2}\pm a_{3})=a{_{1}}^{2}Var(X_{1})+ a{_{2}}^{2}Var(X_{2})\]

If X and Y are random variables following a normal distribution and Z=aX+bY+c then Z also follows a normal distribution.

For any distribution, if \(E(X)=\mu\), \(Var(X)=\sigma ^{2}\) and \(n\geqslant 30\), then the approximate distributions of the sum and the mean are given by:
\[\sum_{i=1}^{n}X_{i}\sim N(n\mu,n\sigma ^{2})\]\[\bar{X}\sim N(\mu,\frac{\sigma^{2}}{n})\]

The cumulative distribution function gives the probability of the random variable taking a value less than or equal to x.

For a discrete distribution with probability mass function \(P(X = x)\):
\[P(X\leq x)=\sum_{i=-\infty }^{i=x}p_{i}\]

For a continuous distribution with pdf \(f(x)\):
\[P(X\leq x)= F(x)=\int_{-\infty }^{x}f(x)dx\]

If X has cdf \(F(x)\) for \(a< x< b\) and \(Y=g(X)\) (where \(g(X)\) is a 1-to-1 function) then the pdf of Y, \(h(y)\), is given by:
1) Relating \(H(y)\) to \(F(g^{-1}(y))\) by rearranging the inequality in \(P(Y\leq X)=P(g(X)\leq y)\).
2) Differentiating \(H(y)\) with respect to y.
3) Writing the domain of \(h(y)\) by solving the inequality \(a< g^{-1}(y)< b\)

why use pdf, for a continuous random variable, the probability can only be in a range (area under the curve)

For a continuous variable it does not matter whether you use strict inequalities (a < × < b) or inclusive inequalities (a ≤ × ≤ b).

A probability density function will often have different rules over different domains. If a probability density function is only defined over a particular domain you may assume that it is zero everywhere else.

Median

Expectation (mean)

It will be not necessary where dt=0 for the x to be the mode of the function.

t-distribution:
\[\frac{\bar{X}-\mu}{S_{n-1}/{\sqrt n}}\sim t_{n-1}\]

Width of confidence interval:
\[2z\frac{\sigma}{\sqrt n}\]

Confidence interval for a mean with unknown variance:
\[\bar{x}\pm t\frac{S_{n-1}}{\sqrt n}\]

General procedure for hypothesis testing:

1. Define variables
2. State hypothesis \[H_0 \ ,\ H_1\]
3. State significance level
4. Decide which type of test statistic
5. State distribution assuming that null hypothesis is true
6. Calculate test statistic
7. Conclusion (state whether the statistic is sufficiently unlikely by using p-value or critical region)

One-tailed and two-tailed test

Error in hypothesis testing:

PDF
\[^{x-1}C_{r-1}\,=p^{r} q^{x-r}\]
where x= r, r+1, r+2, ...

X= Number of trials up to and including r successes

Parameter
X~ NB(r,p)

Expectation
E(X)= r/p

Variance
Var(X)= rq/(p^2)

an accurate statistic that's used to approximate a population parameter