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4.2 - Using the TI-83 to obtain the Mean and Standard Deviation for a Probability
Distribution – Identifying Usual/unusual results with the range rule of thumb, or with the
probability rule
In section 4.2, we were given the following formulas to compute the mean and standard deviation
for a probability distribution.
  [ x  P( x)]
Mean
Standard Deviation

[ x
2
 P( x)]   2
Where x are the possible values of the random variable, and P(x) are the corresponding
probabilities.
Use the calculator to find the mean and the standard deviation of a probability distribution
Enter x into L1
Enter the probabilities into L2
Press STAT
Arrow right to CALC
Select 1: 1-Var Stats L1,L2
Press ENTER
Identifying Unusual Results with the Range Rule of Thumb
According to the range rule of thumb, most values should lie within 2 standard deviations from the
mean. We can therefore identify “unusual” values by determining if they lie outside these limits:
Minimum usual value = μ - 2σ,
Maximum usual value = μ + 2σ
Identifying Unusual Results with Probabilities
Rare Event Rule
If, under a given assumption the probability of a particular observed event is extremely small, we
conclude that the assumption is probably not correct.
Unusually high: x successes among n trials is an unusually high number of successes if P(x or
more) is very small (such as 0.05 or less).
Unusually low: x successes among n trials is an unusually low number of successes if P(x or fewer)
is very small (such as 0.05 or less).
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Example 1:
When randomly selecting jail inmates convicted of DWI (driving while intoxicated), the
probability distribution for the number x of prior DWI sentences is as described in the
accompanying table (based on data from the U.S. Department of Justice).
a) Find the mean and standard deviation of this distribution.
x
0
1
2
3
P(x)
0.512
0.411
0.032
0.045
b) Use the range rule of thumb to determine usual and unusual values?
c) Identify usual and unusual results using the probability rule.
Example 2:
You are taking a multiple choice test with 4 questions, each of which has 5 choices with
only one choice being correct. Assume you are guessing at each answer. The probability
distribution for the number of correct answers is given below.
a) Find the mean and standard deviation of this distribution.
x
0
1
2
3
4
P(x)
0.4096
0.4096
0.1536
0.0256
0.016
b) Use the range rule of thumb to determine usual and unusual values?
c) Identify usual and unusual results using the probability rule.
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4.3 – Using the TI-83 to Find Factorials, Binomial Coefficients and Binomial Probabilities
Here is the binomial probability formula:
P( x) 
n!
 p x  q n x
(n  x)! x!
Using the notation of the binomial coefficient: n C x , the formula can be written as:
P( x) n Cx  p x  q n x or P( x)   nx   p x  q n x
where x is the number of successes in n trials, p is the probability of success in any one trial, and q
is the probability of failure in any one trial. (q = 1 – p)
1) Evaluate factorials with the calculator:
Type number
Press MATH
Arrow left to PRB
Select 4:!
Press ENTER
Examples:
a) Find 9!
9! = 9 * 8 *… * 3 * 2 * 1 =
b) Find 7!
2) Evaluate binomial coefficients with the calculator:
Type n (number of trials)
Press MATH
Arrow left to PRB
Select 3:nCr
Type x
Press ENTER
Examples:
a) Find 12 C3 =
b) Find 9 C5 =
3) Evaluate binomial probabilities with the formula P( x) 
  p
n
x
x
 q n x
a) For a binomial experiment with n = 8 and p = 0.7, find P(x = 2).
b) For a binomial experiment with n = 5 and p = 1/4, find P(x = 3).
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4) Find binomial probabilities with a feature of the calculator
A) To find single probabilities: Use binompdf(n,p,x)
Press 2nd VARS
Select 0:binompdf(
Type n,p,x)
Press ENTER
Examples:
c) For a binomial experiment with n = 8 and p = 0.7, find P(x = 2).
d) For a binomial experiment with n = 5 and p = 1/4, find P(x = 3).
B) To get a list of all the probabilities corresponding to x = 0, 1, 2, …., n:
Use binompdf(n,p) and scroll to the right to read the probabilities
Example 1:
a) For a binomial experiment with n = 3 and p = 1/6, find the probability distribution.
b) Identify usual/unusual results with the probability rule
c) Find the mean and the standard deviation of the random variable
Remember that µ = n p, and σ = sqrt(n*p*(1-p))
d) Identify usual/unusual results with the range rule of thumb
X P(X=x)
Example 2:
a) For a binomial experiment with n = 4 and p = 1/2, find the probability distribution.
b) Identify usual/unusual results with the probability rule
c) Find the mean and the standard deviation of the random variable
Remember that µ = n p, and σ = sqrt(n*p*(1-p))
d) Identify usual/unusual results with the range rule of thumb
X P(X=x)
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C) To calculate cumulative probabilities from 0 to x, use binomcdf(n,p,x)
Press 2nd VARS
Select A:binomcdf(
Type n,p,x)
Press ENTER
Examples:
a) For a binomial experiment with n = 4 and p = 0.25, find the probability of at
most 3 successes.
b) For a binomial experiment with n = 7 and p = 0.4, find the probability of at
most 5 successes.
c) For a binomial experiment with n = 6 and p = 0.2, find the probability of at
least 2 successes.
d) For a binomial experiment with n = 9 and p = 0.95, find the probability of at
least 6 successes.
e) For a binomial experiment with n = 9 and p = 0.35, find the P (3 < x < 7)
f) For a binomial experiment with n = 12 and p = 0.75, find the probability that x is
between 5 and 8 inclusive.
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5.3 – Finding Probabilities in Normal Distributions
Finding probabilities (areas, percentages)
Step 1: Draw the graph shading the area desired. Label the mean
and the specific x-values being considered.
Step 2: Find the z-Score for each x-value involved.
Z-SCORE: z 
x

Note: If you are working with the distribution of sample means, then z 
x
/ n
Step 3: Use table A-2 to find the cumulative left area bounded by z.
Step 4: Answer the problem.
Using the TI-83 to obtain probabilities, percentages, areas
Press 2nd, VARS
Select 2:normalcdf(
Type left endpoint, right endpoint, mean, standard deviation)
Press ENTER
Example:
Heights of women are normally distributive with a mean of 63.6 in. and a standard
deviation of 2.5 in. If we select one woman at random, what is the probability that
a) Her height is between 66.5 in and 71.5 in.? (These are the requirements to be a
Rockette dancer.)
b) Her height is at least 70 inches? (This is the requirement for the Beanstalk Club)
c) Her height is lower than 60 inches?
d) If 25 women are randomly selected what is the probability that the mean height is lower
than 60 inches?
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Finding Values, scores, from Know Areas/Probabilities/ % in a normal distriabution
Step 1: Draw the graph, shade and label the given area, and identify the
x-value being sought.
Step 2: Find the cumulative left area bounded by x.
Step 3: Use table A-2 to find the z-score.
(Go backwards! From the main body of the table to the z-score)
Step 4 Find the score, x, by using the formula
x    z
Using the TI-83 to obtain normal scores, percentiles
Press 2nd, VARS
Select 3:invNorm(
Type total area to the left of the desired value, , )
Press ENTER
Example:
Heights of women are normally distributive with a mean of 63.6 in. and a standard
deviation of 2.5 in. If we exclude the shortest 20% and the tallest 10%, what are the minimum
and maximum heights of the group?
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6.2 – Estimating a Population Proportion – Confidence Intervals
Assumptions
 The sample is a simple random sample. (SRS)
 The conditions for a binomial distribution are satisfied by the sample. That is: there are a
fixed number of trials, the trials are independent, there are two categories of outcomes, and
the probabilities remain constant for each trial. A “trial” would be the examination of each
sample element to see which of the two possibilities it is.
 The normal distribution can be used to approximate the distribution of sample proportions
because np ≥ 5 and nq ≥ 5 are both satisfied. (q = 1 – p)
Procedure for Constructing a Confidence Interval for p
1) Verify that the assumptions are satisfied.
2) Find the critical value z / 2 (from table A-2).
3) Evaluate the margin of error E. E  z / 2
pq
n
4) Find the interval and write it in one of the following forms
pE;
pE p pE;
( p  E, p  E )
Round-Off Rule for Confidence Interval Estimates of p
3 significant digits
Using the TI-83 to Construct Confidence Intervals for p:
STAT>>TESTS choose A:1-propZInt.
Notice: x must be a whole number
If you have to calculate it,
round it to the nearest whole number
Example:
In an October 2003 poll conducted by the Gallup Organization, 684 of 1006 randomly
selected adults aged 18 years old or older stated they think the government should make partial
birth abortions illegal, except in cases necessary to save the life of the mother.
a) Obtain a point estimate for the proportion of adults aged 18 or older who think the government
should make partial birth abortions illegal, except in cases necessary to save the life of the
mother.
b) Verify that the requirements for constructing a confidence interval about p-hat are satisfied.
c) Construct a 98% confidence interval for the proportion of adults aged 18 or older who think
the government should make partial birth abortions illegal, except in cases necessary to save the
life of the mother. Interpret the interval.
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6.3
Estimating a Population Mean: σ Known
Assumptions:



The sample is a simple random sample (All samples of the same size have an equal
chance of being selected.)
The value of the population standard deviation σ is known.
Either or both of these conditions is satisfied:
i) The population is normally distributed, or
ii) n > 30 (The sample has more than 30 values)
Procedure for Constructing a Confidence Interval for μ (with Known σ)
1. Verify that the required assumptions are satisfied.
2. Find the critical value z / 2 (from table A-2).
3. Evaluate the margin of error E.
(E = z / 2

)
n
4. Then using E and the sample mean the confidence interval is:
x  E    x  E or ( x  E , x  E )
Using the TI-83 to Construct Confidence Intervals for μ:
STAT>>TESTS select 7:ZInterval
Example:
A simple random sample of size n is drawn from a population whose population standard
deviation, σ, is known to be 3.8. The sample mean, x-bar, is determined to be 59.2.
a) Compute the 90% confidence interval about µ if the sample size, n, is 45.
b) Compute the 90% confidence interval about µ if the sample size, n, is 55. How does increasing
the sample size affect the margin of error E?
c) Compute the 98% confidence interval about µ if the sample size, n, is 45. Compare the results
to those obtained in part (a). How does increasing the level of confidence affect the size of the
margin of error, E?
d) Can we use the method described here to compute a confidence interval about µ based on the
information given if the sample size is n = 15? Why? If the sample size is 15, what must be true
regarding the population from which the sample was drawn in order to use this method?
e) What is more precise, a longer or a shorter interval?
f) Comment on the relationship between sample size and precision.
g) Comment on the relationship between confidence level and precision.
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6.4
Estimating a Population Mean: σ Not Known
Conditions for Using the Student t Distribution
 σ is unknown, (if σ is known we use z);
 The sample is a simple random sample
 Either or both of these conditions is satisfied:
i) The population is normally distributed, or
ii) n > 30 (The sample has more than 30 values)
Confidence Interval for the Estimate of μ (With σ Not Known)
1. Verify that the required assumptions are satisfied.
2. Find the critical value t / 2 from table A-3 (n -1 degrees of freedom).
3. Evaluate the margin of error E.
(E = t / 2
s
)
n
4. Then using E and the sample mean the confidence interval is:
xE  xE
or
( x  E, x  E )
Using the TI-83 to Construct a Confidence Interval for Estimating μ, (σ Not Known)
STAT>>TESTS select 8:TInterval
Example:
A simple random sample of size n is drawn from a population. If that sample have a mean of 59.2
and a standard deviation of 3.8,
a) Compute the 90% confidence interval about µ if the sample size, n, is 45. Compare with the
answer to part (a) on the previous page. What do you notice?
b) Compute the 90% confidence interval about µ if the sample size, n, is 55. How does increasing
the sample size affect the margin of error, E?
c) Compute the 98% confidence interval about µ if the sample size, n, is 45. Compare the results
to those obtained in part (a). How does increasing the level of confidence affect the size of the
margin of error, E?
d) Can we use the method described here to compute a confidence interval about µ based on the
information given if the sample size is n = 15? Why? If the sample size is 15, what must be true
regarding the population from which the sample was drawn in order to use this method?
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