Statistics How To

Central Limit Theorem: “Between” Probability

Andale — Thu, 26 Jan 2012 20:03:25 +0000

Sample problem: The there are 250 dogs at a dog show who weigh an average of 12 pounds, with a standard deviation of 8 pounds. If 4 dogs are chosen at random, what is the probability they have an average weight of greater than 8 pounds and less than 25 pounds?

Step 1:Identify the parts of the problem. Your question should state:

mean (average or μ) standard deviation (σ) population size
sample size (n)
number associated with “less than” 1
number associated with “greater than” 2

Step 2: Draw a graph. Label the center with the mean. Shade the area between 1 and 2. This step is optional, but it may help you see what you are looking for.

Step 3: Use the following formula to find the z-values.

All this formula is asking you to do is:

a) Subtract the mean (μ in Step 1) from the greater than value (Xbar in Step 1): 25-12=13.
b) Divide the standard deviation (σ in Step 1) by the square root of your sample (n in Step 1): 8/sqrt4=4
c) Divide your result from a by your result from b: 13/4=3.25

Step 4 Use the formula from Step 3 to find the z-values. This time, use Xbar2 from Step 1 (8).

a) Subtract the mean (μ in Step 1) from the greater than value (Xbar in Step 1): 8-12=-4.
b) Divide the standard deviation (σ in Step 1) by the square root of your sample (n in Step 1): 8/sqrt4=4
c) Divide your result from a by your result from b: -4/4= -1

Step 5: Look up the z-value you calculated in Step 3 in the z-table.

Z value of 3.25 corresponds to .4494

Step 6: Look up the z-value you calculated in Step 4 in the z-table.

Z value of 1 corresponds to .3413

Step 7: Add Step 5 and 6 together:

.4494 + .3413 = .7907

Step 8: Convert the decimal in Step 7 to a percentage:

.7907 = 79.07%

How to make a frequency chart and determine frequency

Andale — Fri, 30 Dec 2011 00:51:57 +0000

If you are asked to determine a frequency in statistics, it doesn’t just mean that you should just count out the number of times something occurs.

Step 1: Make a chart for your data. For this example, let’s say you’ve been given a list of twenty blood types for incoming emergency surgery patients:

A, O, A, B, B, AB, B, B, O, A, O, O, O, AB, B, AB, AB, A, O, A

On the horizontal axis, write “frequency (#)” and “percent (%)”. On the vertical axis, write your list of items. In this example, we have four distinct blood types: A, B, AB, and O.

Step 2: Count the number of times each item appears in your data.
In this example, we have:
A appears 5 times
B appears 5 times
O appears 6 times
AB appears 4 times

Write those in the “number” column. This is your frequency.

Step 3:
Use the formula % = (f / n) × 100 to fill in the next column. In this example, n = total amount of items in your data = 20. A appears
5 times (frequency in this formula is just the number of times the item appears). So we have:

(5 / 20) × 100 = 25%

Fill in the rest of the frequency column, changing the ‘f’ for each blood type.

That’s it!

How to Classify a Variable as Quantitative or Qualitative

Andale — Fri, 30 Dec 2011 00:39:24 +0000

In introductory statistics, it’s easy to get confused when classifying a variable or object as quantitative or qualitative. Quantitative means it can be counted, like “number of people per square mile.” Qualitative means it is a description, like “brown dog fur.” How to classify those variables:

Step 1: Think of a category for the items, like “car models” or “types of potato” or “feather colors” or “numbers” or “number of widgets sold.” The name of the category is not important.

Step 2: Rank or order the items in your category. Some examples of items that can be ordered are: number of computers sold in a month, students’ GPAs or bank account balances. Anything with numbers or amounts can be ranked or ordered. If you find it impossible to rank or order your item, you have a qualitative item. Examples of qualitative items are “car models,” “types of potato,” “Shakespeare quotes.”

Step 3: Make sure you haven’t added information. For example, you could rank car models by popularity or expense, but popularity and expense are separate variables from “car model.” If the item is “potatoes,” it’s qualitative. If the item is “number of potatoes sold,” it’s quantitative.

How to Find a Critical Value in Ten Seconds (Two-Tailed Test)

Stephanie — Tue, 06 Dec 2011 16:20:33 +0000

Critical values are used in tests of statistical significance. The alpha level is the maximum probability where you reject the null hypothesis if the null hypothesis is true (in other words, you’re controlling the possibility of Type 1 errors to a certain level). Let’s say you have an alpha level of .05 for this example.

Step 1: Subtract alpha from 1.

1 – .05 = .95

Step 2 :Divide Step 1 by 2 (because we are looking for a two-tailed test)

.95 / 2 = .475

Step 3: Look at your z-table and locate the alpha level in the middle section of the z-table. The fastest way to do this is to use an online z-table and use the find function of your browser (usually CTRL+F). In this example we’re going to look for .475, so go ahead and press CTRL+F, then type in .475.

Step 4:. In this example, you should have found the number .4750. Look to the far left or the row, you’ll see the number 1.9 and look to the top of the column, you’ll see .06. Add them together to get 1.96. That’s the critical value!

Tip: The critical value appears twice in the z table because you’re looking for both a left hand and a right hand tail, so don’t forget to add plus or minus! In our example you’d get ±1.96.

Z-Table: Area to the Left of Curve

Stephanie — Sun, 26 Jun 2011 09:38:47 +0000

Standard Normal Table

Standard Normal (Z) Table

This table shows the area to the left of Z.

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.0

0.5000

0.5040

0.5080

0.5120

0.5160

0.5199

0.5239

0.5279

0.5319

0.5359

0.1

0.5398

0.5438

0.5478

0.5517

0.5557

0.5596

0.5636

0.5675

0.5714

0.5753

0.2

0.5793

0.5832

0.5871

0.5910

0.5948

0.5987

0.6026

0.6064

0.6103

0.6141

0.3

0.6179

0.6217

0.6255

0.6293

0.6331

0.6368

0.6406

0.6443

0.6480

0.6517

0.4

0.6554

0.6591

0.6628

0.6664

0.6700

0.6736

0.6772

0.6808

0.6844

0.6879

0.5

0.6915

0.6950

0.6985

0.7019

0.7054

0.7088

0.7123

0.7157

0.7190

0.7224

0.6

0.7257

0.7291

0.7324

0.7357

0.7389

0.7422

0.7454

0.7486

0.7517

0.7549

0.7

0.7580

0.7611

0.7642

0.7673

0.7704

0.7734

0.7764

0.7794

0.7823

0.7852

0.8

0.7881

0.7910

0.7939

0.7967

0.7995

0.8023

0.8051

0.8078

0.8106

0.8133

0.9

0.8159

0.8186

0.8212

0.8238

0.8264

0.8289

0.8315

0.8340

0.8365

0.8389

1.0

0.8413

0.8438

0.8461

0.8485

0.8508

0.8531

0.8554

0.8577

0.8599

0.8621

1.1

0.8643

0.8665

0.8686

0.8708

0.8729

0.8749

0.8770

0.8790

0.8810

0.8830

1.2

0.8849

0.8869

0.8888

0.8907

0.8925

0.8944

0.8962

0.8980

0.8997

0.9015

1.3

0.9032

0.9049

0.9066

0.9082

0.9099

0.9115

0.9131

0.9147

0.9162

0.9177

1.4

0.9192

0.9207

0.9222

0.9236

0.9251

0.9265

0.9279

0.9292

0.9306

0.9319

1.5

0.9332

0.9345

0.9357

0.9370

0.9382

0.9394

0.9406

0.9418

0.9429

0.9441

1.6

0.9452

0.9463

0.9474

0.9484

0.9495

0.9505

0.9515

0.9525

0.9535

0.9545

1.7

0.9554

0.9564

0.9573

0.9582

0.9591

0.9599

0.9608

0.9616

0.9625

0.9633

1.8

0.9641

0.9649

0.9656

0.9664

0.9671

0.9678

0.9686

0.9693

0.9699

0.9706

1.9

0.9713

0.9719

0.9726

0.9732

0.9738

0.9744

0.9750

0.9756

0.9761

0.9767

2.0

0.9772

0.9778

0.9783

0.9788

0.9793

0.9798

0.9803

0.9808

0.9812

0.9817

2.1

0.9821

0.9826

0.9830

0.9834

0.9838

0.9842

0.9846

0.9850

0.9854

0.9857

2.2

0.9861

0.9864

0.9868

0.9871

0.9875

0.9878

0.9881

0.9884

0.9887

0.9890

2.3

0.9893

0.9896

0.9898

0.9901

0.9904

0.9906

0.9909

0.9911

0.9913

0.9916

2.4

0.9918

0.9920

0.9922

0.9925

0.9927

0.9929

0.9931

0.9932

0.9934

0.9936

2.5

0.9938

0.9940

0.9941

0.9943

0.9945

0.9946

0.9948

0.9949

0.9951

0.9952

2.6

0.9953

0.9955

0.9956

0.9957

0.9959

0.9960

0.9961

0.9962

0.9963

0.9964

2.7

0.9965

0.9966

0.9967

0.9968

0.9969

0.9970

0.9971

0.9972

0.9973

0.9974

2.8

0.9974

0.9975

0.9976

0.9977

0.9978

0.9979

0.9980

0.9981

2.9

0.9981

0.9982

0.9983

0.9984

0.9985

0.9986

3.0

0.9987

0.9988

0.9989

0.9990

What is a Beta Density Function?

Stephanie — Sun, 19 Sep 2010 14:48:55 +0000

A Beta distribution is a family of probability distributions that stretch from 0 to 1 on the number line. The letters α and β define the shape of the curve. The Beta distribution is an excellent way to represent outcomes like probabilities or proportions.

The values of α and β determine the shape of the beta density function. For example, if α < 1 and β < 1, the graph’s shape will be a “U” (see the red plot on the picture above, and if α = 1 and β = 2, the graph is a straight line; If you look at the graph above, the blue line is almost a straight line: that’s because α = 1 and β = 3.

The probability function P(x) and distribution function D(x) for the Beta Distribution are:

It’s unlikely you’ll ever have to use those ugly equations, as most math software simplifies the math for you. For example, in Mathematica you can implement a beta distribution by typing:
BetaDistribution[alpha,beta] and in Microsoft Excel you can use BetaDist(x,shape_a,shape_b,min,max).

The beta distribution is used for many applications, including Bayesian statistics, the Rule of Succession (a famous example being Pierre-Simon Laplace’s treatment of the sunrise problem), and Task duration modeling. the beta distribution is especially suited to project/planning control systems like PERT and CPM because the function is constrained by an interval with a minimum (0) and maximum (1) value.

Tip: don’t get confused by all those betas. In (typical) mathematical tomfoolery, there are three different betas:

In “B(α, β),” Beta is the name of the function in the denominator of the density function.
In “Beta(α, β),” Beta means the name of the probability distribution.
In “β,” Beta is the name of the second parameter in the density function.

References:
Abramowitz, M. and Stegun, I. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 944-945, 1972
Evans, M.; Hastings, N.; and Peacock, B. “Beta Distribution.” Ch. 5 in Statistical Distributions, 3rd ed. New York: Wiley, pp. 34-42, 2000.

How to Use the TI 83 for a Hypothesis Test of a Mean

Stephanie — Thu, 15 Apr 2010 13:38:22 +0000

A hypothesis may or may not be true, and researchers will use data from a random population sample to test the validity of a hypothesis. If the sample data does not agree with the hypothesis, then the hypothesis is rejected. The null hypothesis (H₀) states that the observations occur by chance. The alternate hypothesis (H₁) is just the opposite, and states that the observations were influenced by a non-random cause. You can use the TI 83 calculator for hypothesis testing, but the calculator won’t figure out the null and alternate hypotheses: that’s up to you to read the question and input it into the calculator.

Sample problem: A sample of 200 people has a mean age of 21 with a population standard deviation (σ) of 5. Test the hypothesis that the population mean is 18.9 at α = 0.05.

Step 1: State the null hypothesis. In this case, the null hypothesis is that the population mean is 18.9, so we write:
H₀: μ = 18.9

Step 2: State the alternative hypothesis. We want to know if our sample, which has a mean of 21 instead of 18.9, really is different from the population, therefore our alternate hypothesis:
H₁: μ ≠ 18.9

Step 3: Press Stat then press the right arrow twice to select TESTS.

Step 4: Press 1 to select 1:Z-Test…. Press Enter.

Step 5: Use the right arrow to select Stats.

Step 6: Enter the data from the problem:
μ₀: 18.9
σ: 5
x: 21
n: 200
μ: ≠μ₀

Step 7: Arrow down to Calculate and press Enter. The calculator shows the p-value:
p = 2.87 × 10^-9

This is smaller than our alpha value of .05. That means we should reject the null hypothesis.

How to Get The Mean and Standard Deviation For a Binomial Probability Distribution With a TI 83 Calculator

Stephanie — Thu, 15 Apr 2010 00:29:56 +0000

Binomial probability distributions are used when dealing with two outcomes from a fixed number of independent trials. As a simple example, you might want to figure out the probability of a voter voting yes or no on a certain ballot proposition. P(x) can quickly be calculated on the TI 83 graphing calculator, which has all of the binomial probability tables built into it.

Sample problem: Find the mean and standard deviation for a binomial distribution with n = 5 and p = 0.12.

Mean

Step 1: Multiply n by p.
5 × . 1 2 Enter
.6

Hey, that was easy!

Standard Deviation

Step 1: Subtract p from 1 to find q.
1 - . 1 2 Enter
.88

Step 2: Multiply n times p times q.
5 × . 1 2 × . 8 8 Enter
.528

Step 3: Find the square root of the answer from Step 2.
2nd x² . 5 2 8 Enter
.727 (rounded to 3 decimal places).

How to Find a T distribution on a TI 83

Stephanie — Wed, 14 Apr 2010 23:50:31 +0000

The T distribution (sometimes called Student’s T distribution) is actually a set of distributions, differentiated by the degrees of freedom (df). Take a look at a traditional textbook T table, and you’ll actually find many T tables, which can be a little overwhelming. Instead of poring over tables, you can use a TI 83 graphing calculator to assist you in finding T distribution values.

You might be asked to find the area under a T curve, or (like Z scores), you might be given a certain area and asked to find the T score.

Sample problem: Find the area under a T curve with degrees of freedom 10 for P( 1 ≤ X ≤ 2 ).

Step 1: Press 2nd VARS 5 to select tcdf(.

Step 2: Enter the lower, and upper bounds, and the degrees of freedom: 1 , 2 , 1 0 )

Your screen should now read tcdf(1,2,10)

Step 3: Press ENTER. The answer is .133752549, or about 13.38%.

How to Make a TI 83 box plot

Stephanie — Wed, 14 Apr 2010 23:34:04 +0000

Graphs and charts help us make sense of data, see patterns, and identify trends. Without graphs, it can be difficult to make sense of data. For example, let’s say you have a list of IQ scores for a gifted classroom in a particular elementary school. The IQ scores are: 118, 123, 124, 125, 127, 128, 129, 130, 130, 133, 136, 138, 141, 142, 149, 150, 154. That list doesn’t tell you much about anything. However, with a chart or graph like a box plot (sometimes called a box-and-whisker plot), the data can come to life.

Step 1: Press STAT ENTER, to edit L1.

Step 2: Enter the data from the problem into the list:

1 1 8 ENTER
1 2 3 ENTER
1 2 4 ENTER
1 2 5 ENTER
1 2 7 ENTER
1 2 8 ENTER
1 2 9 ENTER
1 3 0 ENTER
1 3 0 ENTER
1 3 3 ENTER
1 3 6 ENTER
1 3 8 ENTER
1 4 1 ENTER
1 4 2 ENTER
1 4 9 ENTER
1 5 0 ENTER
1 5 4 ENTER

Step 3: Press 2nd Y=, to access the Stat Plot menu.

Step 4: Press ENTER ENTER to turn on Plot1.

Step 5: Arrow down to Type, which has 6 icons to the right of it. Highlight the bottom middle icon, which looks like a syringe with two plungers, and press ENTER to select it.

Step 6: Make sure the XList entry reads “L₁“. If it doesn’t, arrow down to it, Press Clear 2nd 1.

Step 7: Press Graph. You should see your Box plot!

Tip: If when you press Graph, you see the message “Err: Stat”, or you just don’t see a box plot like you expect to, then press Window, and try different settings. Especially try changing the Xscl (X Scale) item to a larger value.