Inferential Statistics Wiki

Write the definition of your choice under its title below.

Confidence Intervals

+ Sampling Distribution

to find percentile for z score - =normdist(z-score,0,1,1)

z-score	percentile
3	0.99865
2	0.97725
1	0.841345
0	0.5
-1	0.158655
-2	0.02275
-3	0.00135

 (Aaron Hines)

+ Confidence Interval:

An interval estimate for an unknown population parameter. This depends on:

• The desired confidence level
• Information that is known about the distribution
• The sample and its size.
• (MIKE DERRICKSON)

        + Confidence Level: 

The percent expression for our confidence that the confidence interval contains the true population parameter. If CL=90%, then in about 90 out of 100 samples the interval estimate will enclose the true populations parameter, but we don't know which ones. (MIKE DERRICKSON and Gary Parker)

to find z score from confidence level  =norminv(alpha/2, 0, 1)  (Aaron Hines)

to find t score from confidence level = t.inv(alpha/2, sample size minus 1)

  

+ Point Estimate

 In a population parameter, its a single value used to estimate the population parameter. (Tanner Heiman)

+ Error Bound for a Population Mean -Laura

to calculate the population mean you must know sigma to use a normal distribution. You must have a confidence level to calculate the z-score. to calculate z score you use the "-norm.inv(left-tail prob, 0,1)" then you can calculate the EBM which is also the margin of error.

margin of error= z score * sigma divided by the square root of n

EMB= z*sigma/sqrt(n) 

+ Student's t-Distribution 

Used for confidence interval and hypothesis tests when `sigma` is unknown. The population distribution should be roughly normal if n is small.

t = t.inv(left-tail prob.,n-1)    NOTE: the function is t-dot-inv()

EBM= t*s/sqrt(n) (Lauren Cagle)

s = standard deviation  =stdev.s(data set) in Excel, or =stdeva(data set) in Google Sheets

+ Error Bound for a Population Proportion- Laura

To calculate the Error bound for a proportion you must first have a confidence level. The confidence level gives you a z score by entering using the norminverse formula. =-normininv(tail, 0,1).

you must also know your p' and q'. To find p' you take the number of successes and divide it by your "n"

p'= number of successes/n.

to find q' you subtract your p' from 1.

  =1-p'

Now you can calculate the error bound.

margin of error= z score* square root of p'*q divided by n

ME= z*sqrt(p'*q'/n)

If there is no p or q given then use .5 as the standard

+ Sample Size Estimate for a Population Mean

 `N=((Z*sigma)/E)^2`   `N=((T*S)/E)^2`    (colin W) -pg. 427

+ Sample Size Estimate for a Population Proportion

   `N=(z^2*p'*q')/(EBP)^2 `   (Devin Sather) -pg. 438

Hypothesis Testing

+ Hypothesis Test

 Based on sample evidence, a procedure for determining whether the hypothesis stated it a reasonable statement and should not be rejected, or is reasonable and should be rejected. (lorasa jodie)

+ Null Hypothesis

 A general statement with no association among groups. (Jared Mathews)

+ Alternative Hypothesis

 Contradictory to the null hypothesis. The sign on this test is how you tell what type of test it is. Either left, right or two tail (Tanner Heiman)

+ Significance Level

 whatever alpha (`alpha)` is equal too. (Jared Mathews)

+ Critical Value

 z- or t- score based on the significance level, `alpha` (Gary Parker)

+ Critical Region

 the region of z- or t-scores outside if the critical z- or t- score. (Gary Parker)

+ P-value

A p-value tells you whether to reject or accept a null hypothesis. If the p-value is less than alpha than that is strong evidence to reject the null hypothesis. If the p-value is greater than alpha than that is strong evidence to accept the null hypothesis.  ( Shyla Davison)

p-value =normdist(x,mean,std,cumulative) (Aaron Hines)

+ Test Statistic

 `z={barx-mu}/{sigma/sqrt{n}}`   for a mean with `sigma` known  or

 `t={barx-mu}/{s/sqrt{n}}`   for a mean with `sigma`  unknown  or

 `z={p'-p}/{sqrt{{p*q} / n}}` for a proportion (Gary Parker)

+ Type I Error

The error you make (`alpha` ) when you reject the null hypothesis, when actually it is true. (Devin Sather)

+ Type II Error

 when the null hypothesis is false and you fail to reject it. (Jared Mathews)

+ Unusual Events

An event is considered unusual if it is sufficiently far from the mean. If the probability of getting a more extreme value then we observe is less than 5% then the event is considered to be unusual.

If the observed value is above the mean and the probability is less than 5%, then we say that the observed value is unusually high.

If the observed value is below the mean and the probability is less than 5%, then we say that the observed value is unusually low. (Peter Parker)

+ Outliers

According to the range rule of thumb, most values should lie within 2 standard deviations of the mean. We can therefore identify “unusual” values by determining if they lie outside these limits. (L.Jodie) 

+ Power of the Test

 The probability that the decision is to reject `H_0` when `H_0`  is false. (Gary Parker)

+ Order for hypothesis testing.

1. State null and alternative hypothesis

2. Identify the test type

3. Calculate test statistic

4. Calculate p-value

5.compare p-value and alpha

6. Reject or not the null

7. Write conclusions (Tanner Heiman)

+ Excel Functions for Means

 P(x<a) = NormDist(a, mu, std, 1) or

=tdist(a, n-1, 1) NOTE: Here "a" must be a t-score (Gary Parker) 

+ Excel Functions for Proportions

Population Proportion - p=x/n  x = number of successes, n = total number

q = 1-p

std of a proportion = sqrt(p*q/n)  (Aaron Hines)

standard error of the distribution 

The standard error for the mean is `sigma/sqrt(n)` 

The standard deviation for a proportion is `sqrt{(p'*q')/n}`   or sqrt(p*q/n)

 to find a new n for proprtion- Laura

n=p*q*(z/desired margin of error)^2

To find a new n for a mean- Laura

n= (z*sigma/desired margin of error)^2

or

n = (t*s/desired margin of error)^2