Download FINITE - UNIT 4 - STATISTICS Ways to Display a Set of Data

FINITE - UNIT 4 - STATISTICS
(Revised 4/16)
Ways to Display a Set of Data:
Frequency Distribution - a list of data and how many times each value occurs
Grouped Frequency Distribution - when the data is grouped in intervals or “modal classes”
Histogram - a bar graph that shows the frequency distribution
Frequency Polygon - a line graph that connects the midpoints of the top of each bar in a histogram.
Measures of Central Tendency: (Locates the center or middle of the data.)
Mean - Determine the sum of the data and divide this sum by the total number of elements in the set.
The mean is represented by x or μ .
Median - Arrange the data in sequential order and then find the middle number.
If there is an even number of data, then find the mean (average) of the two middle numbers.
Mode - The value that occurs most frequently. There may be one mode. OR There may be two or more
data values that occur the same number of times. OR There may be no repeated value.
Measures of Variation: (Measures how far the data is spread out.)
Range - Subtract the smallest value from the largest value in the given set of data.
Standard Deviation - All of the values in the set of data are used to determine this. This can be found by
hand, but it is much easier to use a calculator. The standard deviation is represented by Sx or σ.
TI-Calculator:
When the data set consists of just a list of numbers
STAT → Edit →Type in the data in L1 column (It doesn’t matter what is in L2, since only L1 will be used.)
STAT → CALC → 1-Var Stats, Type in L1 (press 2nd and 1)→Enter. A list of statistical values will be displayed.
x gives the value of the mean
Sx gives the value of the sample standard deviation
For newer TI-84’s: 1-Var Stats→ You will get the menu: List: L1
Freq List: (leave blank)
Calculate (move the cursor down to calculate & press Enter )
When the data set is given with a frequency distribution (interval data)
First, find the midpoint of each interval.
STAT → Edit → Type the midpoints in L1, and type the frequency in L2
STAT → CALC → 1-Var Stats, Type in L1 , L2 then press Enter.
(To type L1, press 2nd and 1; to type L2, press 2nd and 2)
For newer TI-84’s: You will get the menu: List: L1
Freq List: type L2 (by pressing 2nd and 2 , for L2)
Calculate (move cursor down to calculate & press Enter )
THE NORMAL DISTRIBUTION
Normal Curve - A bell-shaped curve that shows the distribution of data. The mean, median and mode all
have the same value, and they occur exactly at the center of the distribution.
There are many normal distributions. But every normal curve has the following properties.
1. Its peak occurs directly above the mean μ.
2. The curve is symmetric about the vertical line through the mean.
3. The curve never touches the x-axis -- it extends indefinitely in both directions.
4. The area under the curve (and above the horizontal axis) is always 1.
(This agrees with the fact that the sum of the probabilities in any distribution is 1.)
It can be shown that a normal distribution is determined by its mean μ and standard deviation σ. A small
standard deviation leads to a tall, narrow curve. A large standard deviation produces a flat, wide curve.
Z-Score: The position of a piece of data, described in terms of the number of standard deviations it is
located from the mean.
x
That is, z = (Piece of data − Mean) / Standard Deviation.
z

A z-score is the number of standard deviations from the mean, so z = 1 corresponds to 1 standard deviation
above the mean, z = −1 corresponds to 1 standard deviation below the mean, and so on.
z-scores →
z = −3
z = −2
z = −1
z =0
z=1
z=2
z=3
Approximately 68% of the data occurs within ± 1 standard deviation of the mean, 95% of the data occurs
within ± 2 standard deviations of the mean, and 99.7% of the data occurs within ± 3 standard deviations.
Area Under a Normal Curve:
The area under the curve between two given values, a and b represents the percentage of data that
occurs between those values. It also represents the probability that an observed data value will be between
those values. (Be sure to pay attention to the directions as to whether it is asking for the probability (which is a
decimal value < 1) or for the percentage value (where you will need to move the decimal point two places to the right).
Standard Normal Distribution (Also called a “z-distribution”)
A “standard normal curve” has a mean of 0 and a standard deviation of 1. You will find the area under a
portion of the curve by using z-scores. This can be done by using the table of z-scores given with each
problem or by using the TI calculator (DISTR → NormalCDF). See examples below.
Normal Distribution
For a “normal curve” you can use the actual values given in the word problem, along with the given mean
and standard deviation (when using DISTR → NormalCDF). You could also convert the given values into
z-scores, and then treat the problem as a standard normal curve. See examples below.
TI-Calculator:
To find the area under the curve from a to b (for either the probability or the percentage)
2nd VARS (Distr) → 2:normalcdf (a, b, μ, σ) (Again, the newer 84’s have a different display than the older ones.)
Where a is the lower value, b is the upper value, μ is the mean, σ is the standard deviation.
If the lower limit is −  , type − 1 EE 99 (scientific notation for −1  1099, which approximates −  ) .
If the upper limit is  , type 1 EE 99 (scientific notation for 1  1099, which approximates  )
When using a “standard normal curve” with the z-scores, μ = 0 and σ = 1.
Examples:
1. A machine produces bolts with an average diameter of 0.25 in. and a standard deviation of 0.02 in.
What is the probability that a bolt will be produced with a diameter greater than 0.3 in.?
Lower limit is 0.3, upper limit is  , mean is 0.25, and stand. dev. is 0.02
Calculator: 2nd VARS (Distr) → 2: normalcdf (0.3, 1 EE 99, 0.25, 0.02) = .0062 (rounded to 4 decimal places)
2. A certain type of light bulb has an average life of 500 hours, with a standard deviation of 100 hours.
The length of life of the bulb can be closely approximated by a normal curve. An amusement park buys
and installs 10,000 such bulbs. Find the total number that can be expected to last between 680 hours
and 780 hours.
2nd VARS (Distr) → 2: normalcdf (680, 780, 500, 100) = .0333750748 (which is the probability).
Then .0333750748  (10,000 bulbs) = 334 bulbs that will last between 680 & 780 hours.
To find the z-score when given the probability to the left of z, and the mean & standard deviation
2nd VARS (Distr) → 3:InvNorm (area to the left of z, μ, σ)
Examples:
1. Find the z-score that indicates 25% of the total area under the standard normal curve is to the right of z.
Note: If 25% is to the right of z, then 75% is to the left of z. For a standard normal curve, μ = 0 and σ = 1
Calculator: 2nd VARS (Distr) → 3: InvNorm (.75, 0, 1) = .67 (rounded to 2 decimal places) → z = .67
2. For a normal distribution with a mean of 60 and a standard deviation of 12, find the value that would
have 30% of the total area to the left of that value.
Calculator: 2nd VARS (Distr) → 3: InvNorm (.30, 60, 12) = 53.7 (rounded to 2 decimal places)
FINITE - UNIT 4 - BINOMIAL PROBABILITY
(Revised 4/16)
A binomial experiment is when
1. The same experiment is repeated a fixed number of times.
2. There are only two possible outcomes, success and failure.
3. The repeated trials are independent, so that the probability of success remains the same for each trial.
To find the probability of x successes in n repeated trials of a binomial experiment, you can use the
following formula or you can use your TI calculator.
Formula: C(n, x) ∙ px ∙ qn – x where C is the combination, n is the # of trials, x is the # of successful trials,
p is the probability of success, q = (1 −p) is the probability of failure
When you are finding the probability of exactly x successes.
TI Calculator: 2nd VARS (Distr) → 0: binomPDF
TI-83: Binompdf (n, p, x) where n = # of trials, p is the probability of success, and x is the # of successes
Newer TI-84: Binompdf Trials: n
P: probability
X value: x
Paste (press Enter to paste these values into Binompdf ( ___, ____, ____)
When you are finding the probability of x or less successes.
TI Calculator: 2nd VARS (Distr) → A: binomCDF (n, p, x)
where n = # of trials, p = probability of success, x = # of successes
Examples:
1. Find the probability that a batter, whose batting average is .320, will have exactly 2 hits at 3 times at bat.
(The two events are either get a hit or not get a hit, and the repeated event is 3 times at bat.)
Calculator: BinomPDF (3, .320, 2) = .208896
or
Formula: C(3, 2) ∙ (.320)2 ∙ (.680)3−2 = .208896
↑(1‒.320)
2. Find the probability that a batter, whose batting average is .320, will have 2 or less hits at 3 times at bat.
2 or less hits → x = 2 (for 2 or less hits)
Calculator: BinomCDF (3, .320, 2) = .967232
3. Find the probability that a batter, whose batting average is .320, will have less than 2 hits at 3 times at bat
Less than 2 hits → He has 0 or 1 hit → x = 1 (for 1 or less hits)
Calculator: BinomCDF (3, .320, 1) = .758336
4. Find the probability that a batter, whose batting average is .320, will have at least 2 hits at 6 times at bat.
At least 2 hits → He has 2, 3, 4, 5 or 6 hits. Remember that BinomCDF only works for x or less events.
Thus, first find the probability that he gets less than 2 hits → 0 or 1 hit → x = 1 (for 1 or less hits)
Calculator: BinomCDF (6, .320, 1) = .378022727
The probability of at least 2 hits = 1 – probability of less than 2 hits. = 1 ‒ .378022727 = .621977273
SUMMARY OF TI CALCULATOR FUNCTIONS
STATISTICS
To find the mean and standard deviation for a set of data:
Single list of data values: STAT → EDIT →Type the data in column L1
Then STAT → CALC → 1-Var Stats L1 (Enter)
Frequency distribution of data: STAT → EDIT →Type the data in column L1 and frequency in L2
Then STAT → CALC → 1-Var Stats L1 , L2 (Enter)
To find the area under a normal curve in order to find the percentage of data or probability:
2nd VARS (Distr) → 2: Normal CDF ( lower limit, upper limit, mean, standard deviation )
To find the z-score when given the area under the normal curve:
2nd VARS (Distr) → 3: InvNorm ( area to the left of z, mean, standard deviation)
BINOMIAL PROBABILITY
To find the probability of exactly x successes:
2nd VARS (Distr) → 0: Binom PDF ( # of trials, probability of success, # of successes)
To find the probability of x or less successes:
2nd VARS (Distr) → A: Binom CDF ( # of trials, probability of success, x or less successes )