Applying Statistics Knowledge to Calculators

• Press 2ND followed by DATA. X01= will appear on the left.
• If data has been entered previously, clear it now by pressing:    2ND   CLR WORK. You will not be reminded to do this in examples provided in the rest of this document.
• Press buttons:    1 ENTER .
• Press ↓ twice and X02= will appear on the left.
• Press buttons:    2   ENTER .
• Press ↓ twice and X03= will appear on the left.
• Press buttons:    3   ENTER .
• Press buttons:  2ND   STAT and then press the ↓ twice until x̄ appears on the left and the mean appears on the right. Answer is 2 for this data.

• Press buttons:   2 × .25 =
• 0.5 appears.
• Press buttons:   + 3 × .1 =
• 0.8 appears (the total of 0.5 plus 0.3).

• Multiply each value of x by the probability of x.
• Sum the products to get the Expected Value (Mean).
• Square each value of x.
• Multiply each result by the probability of x.
• Sum the products for the interim value.
• Square the Expected Value.
• Then subtract the squared Expected Value from the interim value to get Variance.

There are many steps but on the BAII calculator, you can press the + button after each iteration of step 1, accomplishing steps 1 and 2 in one combined step.  For example, using the table below, you would press buttons:  0 × .65 = +   1 × .10 = +   2 × .20 = +   3 × .05 =   to get expected value 0.65.  Steps 3, 4 and 5 can also be done together as one step in the same manner.  Don't use this short cut if your professor wants you to show each product for full marks.

X	Probability	X(Probability)	X2	X2(Probability)
0	.65	0	0	0
1	.10	0.10	1	0.1
2	.20	0.40	4	0.45
3	.05	0.15	9	0.45
Totals:		0.65		1.35

Variance	=  Sum(X2×Probability) − Mean2
	=  1.35 − 0.652
	=  1.35 − 0.4225 = 0.9275

We suggest you check your calculations twice because with so many steps, it is easy to enter one number incorrectly.

• Calculate the mean (average) for your data.
• Subtract the mean from each data item.
• Square each result, which removes any negative signs.
• Add up all the results to get the total.
• The last step is divide the total by the number of data items if this is a population or by the number of data items minus one if this is a sample.

On the BAII calculator, you could use the following method to save time:

• Calculate the mean (average) for your data.
• Press 2ND followed by DATA. X01= will appear on the left.
• Using the table below as an example, you would press 1 − 3 = ENTER to enter the first X value (-2 in this case). Press ↓ ↓ and X02= will appear on the left. Repeat this step for each value of x.
• Press 2ND followed by STAT and then press the ↑ repeatedly until ∑X² appears and the answer is on the right. In the example below it is 8.
• Divide the total by the number of data items if this is a population or by the number of data items minus one if this is a sample.

You should practice doing variance by calculator a few times so you can do each step from memory and with confidence.  Use growingknowing.com to practice questions until you can do it correctly 3 or 4 times in a row.  The following table summarizes these calculations, where mean = x̄ = (1+5+3)/3 = 9/3 = 3 and x_i represents each data item:

xi	xi - x̄	(xi - x̄)2
1	1 - 3 = -2	4
5	5 - 3 = +2	4
3	3 - 3 =   0	0
Totals:	0	8
Variance = s2 = ∑(xi − x̄)2 / (n-1) = 8/(3-1) = 8/2 = 4

Binomial probability P(x) = _nC_xp^x(1-p)^(n-x) where n is the number of trials, x is the number of successes you want, and p is the probability of each trail. 

Example: 

In Excel, you were taught to do Normal Distribution questions directly from data to the probability answers.  With calculators, we must first calculate z using the formula z = (x - mean) / Standard Deviation.  Then use the cumulative normal table to lookup the probability.  The z calculation is an interim step on calculators that cannot be skipped.  Calculators allowed on the CGA test have no button for normal distribution probabilities so you must use a printed table to find z. 

There are small differences in how tables are displayed.  Use the Cumulative normal table (Appendix E in our website) for z values.  You must be careful if your book provides both a CUMULATIVE and a STANDARD normal table.  Excel uses cumulative standard normal values.  We have used the Cumulative table for our manual calculator examples, since the cumulative table is the easiest to understand and use.  The following link will bring up the cumulative normal table in a new window that you can keep open while you continue working on this topic:   Cumulative Areas Under the One-Tailed Normal Curve.

We recommend you review all of the "Manual" calculations in the entire textbook topic "Normal Probability" in the growingknowing.com website.  We recommend you do lots of practice questions.  As with Excel, there are three types of questions; more-than, less-than, and between. Since z values are used in many statistics formulas in our course, we will review each formula below for every topic using z.

In your Excel class you learned how to state the Hypothesis question, find the Test Statistic, decide whether the Decision Rule was positive or negative, and determine whether to Reject a Hypothesis; these skills are exactly the same for calculator and Excel users.

Decision Rule:  Example 1

Decision Rule:  Example 2

Excel does not have a function for Sample Size so Excel students use the same formula as calculator students.  The only extra challenge for calculator users is to find the z value using the Cumulative Normal Table rather than using the =NORM.S.INV function in Excel.  The process is exactly as the same as the 2-tail Hypothesis Decision Rule we showed above.  Always use the 2-tail method when doing Sample Size calculations. 

The following is the formula for sample size:  
Where σ is the Greek symbol for standard deviation and E is the allowable margin of sampling error.

Once you have the z value, you then multiply z by the standard deviation, divide by the error E, and then raise the result to the power of 2.  The only step that is different for calculator versus Excel students for Sample Size calculations (mean or proportion) is calculator students use the Normal Table to find z.

Example:

There is no Excel function for Confidence proportion so Excel and calculator students use the same method once the z value is found.  We use the Normal table to find the z value, the same two-tail method we used in Sample Size and the Decision Rule for Hypothesis Means.  Both Excel and calculator students add the Margin of Error to the mean to find the upper confidence interval and subtract the Margin of Error for the lower confidence interval.  Always use the 2-tail method when doing Confidence or Sample Size calculations.

For Standard Error, Excel and calculator students use the same method because Excel has no function to calculate Standard Error.  Standard Error is the standard deviation divided by the square root of n where n is the sample size.

For Confidence Means, Excel has the =CONFIDENCE function to obtain the Margin of Error.  For calculators, we multiply the Standard Error by the z value to obtain the Margin of Error.  Again, the z value is a 2-tail method shown in Sample Size.

Example:

What is the Confidence Interval where sample is 75, mean = 61, standard deviation (σ) = 8.54, and confidence level of 96%? 

Samples under 30 are small and require the use of t instead of z.  Excel has a =T.INV function to lookup t values while calculator students use a printed t Table.  The following link will bring up the Student's t distribution table in a new window that you can keep open while you continue working on this topic:   Student's t Distribution Table.  Both Excel and calculator students use the concept of degrees of freedom to find t.  Degrees of freedom (df)  =  n - 1.

For Confidence questions, we use a 2-tail t table.  Using the Student t table in growingknowing.com, you find t as follows:  On the left side of the table you find the degrees of freedom column.  On the top of the table, you locate your confidence level in the Two-Tail row, then scan down the confidence level column to read the t value that corresponds with the appropriate degrees of freedom.  Once you have the t value, the calculations for small sample confidence are the same as large sample confidence except you use t instead of z.

Be aware t tables can be confusing if you switch textbooks because each book labels the t table columns differently.  All books use degrees of freedom to label the rows, but some tables label the columns with confidence level, some use 1 or 2 tail, and some label using alpha instead of confidence level.  We avoid confusion at growingknowing.com by using all these labels so whatever book or label you prefer, our table is easy to use.  If your advanced statistics course uses a different column label than the one you prefer, you can use growingknowing.com to see the equivalent label name.  For example, the Keller textbook is popular for advanced statistics courses, and they label their t table using a 1-tail alpha value.  So a Keller t_.05 is the alpha value for a 1-tail test, and by looking at the growingknowing.com t table, you can see if you double the 1-tail alpha value, you can use the same column for a two-tail test.

For small sample hypothesis questions, the only difference is the Decision Rule that uses a t table instead of z.  For hypothesis, you need to be sure you read the correct t table one-tail or two-tail column.  Other than using the t table instead of an Excel function, all calculations are the same as large sample hypothesis above.