HP 35s User Manual

							 
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HP 35s  Averages and standard deviations 
 
hp calculators - 3 - HP 35s  Averages and standard deviations - Version 1.0 
 Figure 2 
 
 To find the value two standard deviations above and below the average, press the following: 
 
 In RPN mode: 
 
 %&15#$67%879 
 
 In algebraic mode: 
 
 %&516#$ (computes the above value) 
 #$915%& (computes the below value) 
 
 Figure 3 
 
Answer: The average sales price is $200,355 and the sample standard deviation is $11,189. Within two  
 standard deviations on either side of this average, in this case between $177,977 and $222,733, 95%  
 of all home sales prices should fall. If a home were to sell for $240,000 in this area, it would be an  
 unusual event. Figure 3 indicates the display in RPN mode. 
 
Example 2: The table below indicates the high and low temperatures for a winter week in Fairbanks, Alaska.  
 What are the average high and low temperatures for this week? 
 
  Sun Mon Tues Wed Thurs Fri Sat 
 High 6 11 14 12 5 -2 -9 
 Low -22 -17 -15 -9 -24 -29 -35 
 
Solution: Be sure to clear the statistics / summation memories before starting the problem. 
 
 %)*+
 
 In either RPN or algebraic mode: 
 
 11:3!,4:,,!,0:,*! 
 -:,1!1*:0!1-:1:!+
+20:-:!+
 
 To find the average high temperature, press: #$. Figure 4 displays the menu shown. 
 
 Figure 4 
 
 To find the average low temperature, press: (. Figure 5 displays the menu shown.   
						

							 
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HP 35s  Averages and standard deviations 
 
hp calculators - 4 - HP 35s  Averages and standard deviations - Version 1.0 
   Figure 5 
  
Answer: The average high temperature that week was 5.29 degrees and the average low temperature was  
 –21.57 degrees. 
 
Example 3: John has bought gas on his driving trip at four gasoline stations as follows: 15 gallons at $1.56 per gallon,  
 7 gallons at $1.64 per gallon, 10 gallons at $1.70 per gallon and 17 gallons at $1.58 per gallon. What is the  
 average price of the gasoline purchased? 
 
Solution: The HP 35s has a weighted average mean calculation built-in that will solve this problem easily. Be sure  
 to clear the statistics / summation memories before starting the problem. 
 
 %)*+
 
 In RPN or algebraic mode: 
 
 ,0,;03!4,;3*!,/,;4! 
 ,4,;0.!+
+
+To find the weighted average price of gasoline purchased, press: #$((. Figure 6 displays  
 the menu shown. 
 
 Figure 6 
 
Answer: The average price per gallon John has paid on his trip is slightly less than $1.61. 
   
						

							 
 
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HP 35s  Probability – Rearranging items 
 
 
 
 
Rearranging items 
 
Practice solving problems involving factorials,  
permutations, and combinations 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
   
						

							 
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HP 35s  Probability – Rearranging items 
 
hp calculators - 2 - HP 35s  Probability – Rearranging items - Version 1.0 
Rearranging items 
 
There are a great number of applications that involve determining the number of ways a group of items can be 
rearranged. The factorial function, accessed by pressing ! (which is the right-shifted function of the # key) on 
the 35s, will determine the number of ways you can rearrange the total number of items in a group. Note that the 35s will 
interpret the factorial function as the gamma function if the argument for the function is a non-integer real number. The 
permutation function, accessed by pressing !$ (which is the right-shifted function of the % key), will return the 
number of ways you can select a subgroup of a specified number of items from a larger group, where the order of each 
of the items in the subgroup is important. The combination function, accessed by pressing & (which is the left-
shifted function of the % key), will return the number of ways you can select a subgroup of a specified number of items 
from a larger group, where the order of each of the items in the subgroup is not important.  
 
To see the difference between permutations and combinations, consider the set of three items A, B, and C. If we select a 
subgroup of 2 items, we could select AC and CA as two possible subgroups. These would be counted as different 
subgroups if computing the number of permutations, but only as one subgroup if computing the number of combinations. 
Note that the factorial function operates the same in algebraic mode as it does in RPN mode. The number is keyed in 
and then the factorial function is selected from the keyboard. For permutations and combinations in RPN mode, the two 
numbers must be entered into the first two levels of the stack and then the function is selected from the keyboard. In 
algebraic mode, permutations and combinations require the function to be selected, then the first number to be keyed in 
and then the second number keyed in followed separated from the first by a 35s supplied comma followed by pressing 
the ( key to evaluate the function. 
 
Factorials show up throughout mathematics and statistics. Permutations and combinations show up in many discrete 
probability distribution calculations, such as the binomial and hypergeometric distributions.  
Practice solving problems involving factorials, permutations, and combinations  
Example 1: How many different ways could 4 people be seated at a table?  
Solution: In RPN or algebraic mode: )! 
 
 Figure 1  
Answer: 24.  Figure 1 shows the display assuming RPN mode.  
Example 2: How many different hands of 5 cards could be dealt from a standard deck of 52 cards? Assume the  
 order of the cards in the hand does not matter. 
 
Solution: Since the order of the cards in the hand does not matter, the problem is solved as a Combination.  
 In RPN mode: *+(*&  
 In algebraic mode: &*+,*( 
 
 Figure 2 
   
						

							 
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HP 35s  Probability – Rearranging items 
 
hp calculators - 3 - HP 35s  Probability – Rearranging items - Version 1.0 
Answer: 2,598,960 different hands. Figure 2 shows the display assuming algebraic mode. 
 
Example 3: John has had a difficult week at work and is standing in front of the doughnut display at the local  
 grocery store. He is trying to determine the number of ways he can fill his bag with his 5 doughnuts  
 from the 20 varieties in the display case. He considers the order in which the doughnuts are placed into  
 the bag to be unimportant. How many different ways can he put them in his bag? 
 
Solution: Since the order in which the doughnuts are placed in the bag does not matter, the problem is solved  
 as a combination. 
 
 In RPN mode: +-(*& 
 
 In algebraic mode: &+-,*( 
 
 Figure 3 
 
Answer: 15,504 different ways. Figure 3 shows the display assuming RPN mode. 
 
Example 4: John has had a difficult week at work and is standing in front of the doughnut display at the local  
 grocery store. He is trying to determine the number of ways he can fill his bag with his 5 doughnuts 
 from the 20 varieties in the display case. He considers the order in which the doughnuts are placed into  
 the bag to be quite important. How many different ways can he put them in his bag? 
 
Solution: Since the order in which the doughnuts are placed in the bag matters, the problem is solved  
 as a permutation. 
 
 In RPN mode: +-(*!$ 
 
 In algebraic mode: !$+-,*( 
 
 Figure 4 
 
Answer: 1,860,480 different ways. John may be in front of the display case for some time. Figure 4 shows the  
 display assuming algebraic mode. 
 
Example 5: If you flip a coin 10 times, what is the probability that it comes up tails exactly 4 times?  
 
Solution: This is an example of the binomial probability distribution. The formula to find the answer is given by:   
						

							 
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HP 35s  Probability – Rearranging items 
 
hp calculators - 4 - HP 35s  Probability – Rearranging items - Version 1.0 
   Figure 5 
 
 where P(X) is the probability of having X successes observed, nCx is the combination of n items  
 taken x at a time, and p is the probability of a success on each trial. 
 
 In RPN mode:  .-()&-/*()0%1
11.(-/*2.-()20% 
 
 In algebraic mode: &.-,),%-/*0)%4.2-/*1
11,04.-2)(1
 
 Figure 6 
 
Answer: If you flip a coin 10 times, there is a 20.51% chance of seeing heads 4 times. Figure 6 indicates the 
 display if solved in algebraic mode. 
   
						

							 
 
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HP 35s  Normal distribution applications 
 
 
 
 
The normal distribution 
 
Entering the normal distribution program 
 
Practice solving problems involving  
the normal distribution 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
   
						

							 
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HP 35s  Normal distribution applications 
 
hp calculators - 2 - HP 35s  Normal distribution applications - Version 1.0 
The normal distribution  
The normal distribution is frequently used to model the behavior of random variation about a mean. This model assumes 
that the sample distribution is symmetric about the mean, M, with a standard deviation, S, and generates the shape of 
the familiar bell curve. A standardized normal distribution has a mean of 0 and a standard deviation of 1. This results in 
the familiar Z value used in normal distribution problems to signify the number of standard deviations above or below the 
mean a particular observation falls. It is computed using the formula shown below.  
 !
#$XZ    Figure 1 
 
where X is the observation,  is the mean and ! is the standard deviation. Z is often called a Z-score.  
Entering the normal distribution program  
Solving problems involving the normal distribution requires the entry of the program below into the HP 35s calculator. 
This program can be found in chapter 16 of the HP 35s RPN/ALG Scientific Calculator Owner’s Manual. 
 
Given a value x, this program calculates the probability that a random selection from the sample data will have a higher 
value. This is known as the upper tail area, Q(x). This program also provides the inverse: given a value Q(x), the 
program calculates the corresponding value x. This program uses the built–in integration feature of the HP 35s to 
integrate the equation of the normal frequency curve. The inverse is obtained using Newtons method to iteratively 
search for a value of x which yields the given probability Q(x). The program as listed will work in RPN mode only and that 
mode is assumed throughout this training aid. 
 
After each labels section in the program listing below, the checksum and length are displayed in parentheses to the right 
of the page. These should match what you see on your HP 35s if you have entered the program correctly. To see each 
checksum once you have completely entered the program below, while still in program mode, press !#$
%!&. Pressing  will scroll down through each label in the program showing its length. To see each 
checksum, press!& when each label is displayed. For example, in the listing below, the (70BF – 26) indicates a 
checksum of 70BF and a length of 26 bytes.  
 
In RPN mode, press the following keys to prepare for entry of the program (WARNING: Doing this will erase all of 
program memory): 
 
()(*+,%$
 
Once this is done, key in the following program: 
 
(-./(01!213(0.!2.!4 (70BF – 26) 
(-5!2678%(08!98:5% (042A – 18) 
(-;!28
						

							 
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HP 35s  Normal distribution applications 
 
hp calculators - 3 - HP 35s  Normal distribution applications - Version 1.0 
Practice solving problems involving the normal distribution 
 
Example 1: Find Q(x) for a Z value of +1. Make sure the HP 35s is in RPN mode.  
Solution: With the input value given as a Z-score, were dealing with the standardized normal distribution having a 
mean of 0 and a standard deviation of 1. Press 9! to enter RPN mode. 
 
 In RPN mode:  7.$
 
 Figure 2 
 
 Since we are dealing with a standardized normal distribution, the mean should stay equal to 0. 
$
 In RPN mode:  P$ 
 Figure 3 
  
 Since we are dealing with a standardized normal distribution, the standard deviation is equal to 1. 
$
 In RPN mode:  P$
 
 Now, calculate Q(x) for an x value of 1 by pressing: 
 
 In RPN mode:  75$
 
 Figure 4 
 
 In RPN mode:  3P$
 
 Figure 5 
 
Answer: The upper tail probability for the standardized normal distribution with a value of x equal to +1 is 0.1587. 
This means that only 15.87% of all values would be larger than a Z-score of +1.   
Example 2: Find Q(x) for a Z value of -1. Make sure the HP 35s is in RPN mode.  
Solution: With the input value given as a Z-score, were dealing with the standardized normal distribution having a 
mean of 0 and a standard deviation of 1. Press 9! to enter RPN mode.   
						

							 
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HP 35s  Normal distribution applications 
 
hp calculators - 4 - HP 35s  Normal distribution applications - Version 1.0 
In RPN mode:  7.$
 
 Figure 6 
 
 Since we are dealing with a standardized normal distribution, the mean should stay equal to 0. 
$
 In RPN mode:  P$ 
 Figure 7 
  
 Since we are dealing with a standardized normal distribution, the standard deviation is 1. 
$
 In RPN mode:  P$
 
 Now, calculate Q(x) for an x value of –1 by pressing: 
 
 In RPN mode:  75 (Note: If example 2 is done right after example 1, then figure 4 will show 
a prompt of 1 rather than the zero shown)$
 
 Figure 8 
 
 In RPN mode:  3LP$
 
 Figure 9 
 
Answer: The upper tail probability for the standardized normal distribution with a value of x equal to -1 is 0.8413. 
This means that 84.13% of all values would be larger than a Z-score of –1. Conversely, 15.87% of all 
values would be smaller than a Z-score of –1.   
Example 3: The average number of claims processed per hour by an insurance adjuster is 15 with a standard deviation 
of 4 and follows the normal distribution. If an adjuster processes 20 claims per hour, what percentage of 
adjusters is this person performing faster than?  
Solution: This is a normal distribution problem where the input is not standardized. Press 9! to enter RPN 
mode. Then execute label S and enter the mean and standard deviation. 
 
 In RPN mode:  7.3MPQP$