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CBC Mathematics Division: Math 1442 Elementary Statistics
Exam Formula Sheets
Test 1: Ch 1-3
⦿ Relative Frequency(RF) =
Class Frequency
Sum of All Frequencies
⦿ Percentage Frequency = RF × 100%
⦿ Angle Frequency = RF × 360°
Upper limit of one class + Lower limit of next class
2
Maximum value − mininum value
⦿ Class width =
Number of classes
Lower Class Limit + Upper Class Limit
⦿ Class Midpoint =
2
⦿ Class Boundaries =
𝑁
𝑋1 + 𝑋2 + ⋯ + 𝑋𝑁 1
⦿ Population mean 𝜇 =
= ∑ 𝑋𝐼
𝑁
𝑁
𝐼=1
𝑛
⦿ Sample mean 𝑥̅ =
𝑥1 + 𝑥2 + ⋯ + 𝑥𝑛 1
= ∑ 𝑥𝑖
𝑛
𝑛
𝑖=1
Finding Median
Arrange the data in ascending order
(Smaller to bigger number)
The value at
𝑛+1 𝑡ℎ
2
position
Is the number
of data (𝑛)
Odd or Even
Odd
Skewed to the left
(Negative skewed)
Mean = Median = Mode
The average of the values of
𝑛 𝑡ℎ
2
Normal
(Bell-shaped)
Mean ≤ Median ≤ Mode
⦿ Midrange =
Even
and
𝑛
2
+1
𝑡ℎ
position
Skewed to the right
(Positve skewed)
Mode ≤ Median ≤ Mean
Largest value + Smallest value
2
⦿ Range = (Maximum value) − (Minimum value)
𝐶ℎ𝑒𝑜𝑛-𝑆𝑖𝑔 𝐿𝑒𝑒
Mathematics Division-Coastal Bend College
e-mail:[email protected]
Page 1
CBC Mathematics Division: Math 1442 Elementary Statistics
Exam Formula Sheets
𝑛
⦿ Mean Absolute Deviation (MAD) =
1
∑|𝑥𝑖 − 𝑥̅ |,
𝑛
𝑖=1
𝑁
⦿ Population Variance 𝜎 2 =
1
∑(𝑋𝐼 − 𝜇)2 ,
𝑁
𝑁
where 𝜇 =
𝐼=1
⦿ Population Standard Deviation 𝜎 =
1
∑ 𝑋𝐼
𝑁
𝐼=1
√𝜎 2
(∑ 𝑥𝑖 )2
𝑛
∑ 𝑥𝑖2 −
1
𝑛
⦿ Sample Variance 𝑠 2 =
∑(𝑥𝑖 − 𝑥̅ )2 or
𝑛−1
𝑛−1
or
𝑖=1
⦿ Sample Standard Deviation s = √𝑠 2
𝑠
⦿ Coefficient of variation = CV = ∙ 100%,
𝑥̅
𝜎
⦿ Coefficient of variation = CV = ∙ 100%,
𝜇
𝑛(∑ 𝑥𝑖2 ) − (∑ 𝑥𝑖 )2
𝑛(𝑛 − 1)
for a sample.
for a population.
⦿ The empirical rule for normal distribution is defined as
1.
2.
3.
68% of data values fall in 1 standard deviation (𝜎) of the mean.
= 68% of data values fall in the interval (𝜇 − 𝜎, 𝜇 + 𝜎)
95% of data values fall in 2 standard deviation (2𝜎) of the mean.
= 95% of data values fall in the interval (𝜇 − 2𝜎, 𝜇 + 2𝜎)
99.7% of data values fall in 3 standard deviation (3𝜎) of the mean.
= 99.7% of data values fall in the interval (𝜇 − 3𝜎, 𝜇 + 3𝜎)
68%
95%
99.7%
𝜇 − 3𝜎 𝜇 − 2𝜎 𝜇 − 𝜎
𝜇
𝜇+𝜎
𝜇 + 2𝜎 𝜇 + 3𝜎
⦿ 𝜇 − 2𝜎 ≤ Usual values ≤ 𝜇 + 2𝜎
⦿ Unusual values < 𝜇 − 2𝜎 or Unusual values > 𝜇 + 2𝜎
⦿ Chebyshev’s theorem: 1 −
1
𝑘2
𝑥𝑖 − 𝜇
𝜎
𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠 𝑖𝑛 𝑒𝑣𝑒𝑛𝑡 𝐸
⦿ 𝑃(𝐸) =
𝑇𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠
⦿ 𝑧­score: 𝑧 =
⦿ Complement of event 𝐸 is denoted by 𝐸 𝑐 or 𝐸̅ : 𝑃(𝐸) + 𝑃(𝐸̅ ) = 1
⦿ Multiplication Rule
𝑃(𝐴 ∩ 𝐵) = 𝑃(𝐴) ∙ 𝑃(𝐵|𝐴) for Dependent Events
𝑃(𝐴 ∩ 𝐵) = 𝑃(𝐴) ∙ 𝑃(𝐵) for Independent Events
𝐶ℎ𝑒𝑜𝑛-𝑆𝑖𝑔 𝐿𝑒𝑒
Mathematics Division-Coastal Bend College
e-mail:[email protected]
Page 2
CBC Mathematics Division: Math 1442 Elementary Statistics
Exam Formula Sheets
⦿ Additional Rule
𝑃(𝐴 ∪ 𝐵) = 𝑃(𝐴) + 𝑃(𝐵) − 𝑃(𝐴 ∩ 𝐵)
𝑃(𝐴 ∪ 𝐵 ∪ 𝐶) = 𝑃(𝐴) + 𝑃(𝐵) + 𝑃(𝐶) − 𝑃(𝐴 ∩ 𝐵) − 𝑃(𝐴 ∩ 𝐶) − 𝑃(𝐵 ∩ 𝐶) + 𝑃(𝐴 ∩ 𝐵 ∩ 𝐶)
𝑃(𝐴 ∪ 𝐵) = 𝑃(𝐴) + 𝑃(𝐵) for mutually exclusive events
⦿ Factorial Rule: The collection of 𝑛 distinct objects can be arranged in order 𝑛! = 𝑛 ∙ (𝑛 − 1) ⋯ 3 ∙ 2 ∙ 1
⦿ Permutations and Combinations
Simple Random Samples
Yes
Are samples replaced
for next selection?
Number of WR smaples is
𝑵𝒓
No
Are rearrangements of
the same elements
considered to be same?
Yes
𝐂𝐨𝐦𝐛𝐢𝐧𝐚𝐭𝐢𝐨𝐧
Number of WOR smaples is
𝑁!
𝑁
= 𝑁.𝐶𝑟 =
𝑟
(𝑁 − 𝑟)! 𝑟!
No
Permutations
Are elements in the
sample identical?
Yes
𝐏𝐞𝐫𝐦𝐮𝐭𝐚𝐭𝐢𝐨𝐧𝐬
Number of WOR smaples is
𝑛!
𝑛1 ! 𝑛2 ! ⋯ 𝑛𝑘 !
No
𝐏𝐞𝐫𝐦𝐮𝐭𝐚𝐭𝐢𝐨𝐧𝐬
Number of WOR smaples is 𝑛.𝑃𝑟. =
𝑛!
(𝑛 − 𝑟)!
Test 2: Ch 4-5
𝐸(𝑥) = 𝜇 = ∑ 𝑥𝑖 𝑃(𝑥𝑖 );
𝜎 2 = ∑(𝑥𝑖 − 𝜇)2 𝑃(𝑥𝑖 );
𝜎 = 𝜎 2 = √∑(𝑥𝑖 − 𝜇)2 𝑃(𝑥𝑖 )
⦿ Binomial Distribution
𝑃(𝑥) = 𝑛1𝐶𝑥 𝑝𝑥 𝑞 𝑛−𝑥 =
𝑛!
𝑝 𝑥 𝑞 𝑛−𝑥 ; 𝜇 = 𝑛𝑝; 𝜎 2 = 𝑛𝑝𝑞; 𝜎 = √𝑛𝑝𝑞
(𝑛 − 𝑥)! 𝑥!
⦿ Geometric Distribution: 𝑃(𝑥) = 𝑝𝑞 𝑥−1
𝐶ℎ𝑒𝑜𝑛-𝑆𝑖𝑔 𝐿𝑒𝑒
Mathematics Division-Coastal Bend College
e-mail:[email protected]
Page 3
CBC Mathematics Division: Math 1442 Elementary Statistics
Exam Formula Sheets
⦿ Poisson Distribution:
𝑃(𝑥) =
⦿ 𝑧­score: 𝑧 =
𝜆𝑥 𝑒 −𝜆
;
𝑥!
𝑀𝑒𝑎𝑛 = 𝑉𝑎𝑟𝑖𝑎𝑛𝑐𝑒 = 𝜆 = 𝑛𝑝
𝑥𝑖 − 𝜇
𝜎
Population
⦿ Central Limit Theorem
1. Mean of all values of 𝑥̅ = 𝜇𝑥̅ = 𝜇
2. Standard deviation of all values of 𝑥̅ = 𝜎𝑥̅ =
𝑧=
𝜎
√𝑛
Normally
distributed?
𝑥̅ − 𝜇
𝜎
√𝑛
No
𝑛 > 30
No
Other Methods
⦿ Normal Distribution to Approximate Binomial Probabilities
If 𝑛𝑝 ≥ 5 and 𝑛𝑞 ≥ 5 for a Binomial Probability, then
𝜇 = 𝑛𝑝;
𝜎 = √𝑛𝑝𝑞;
𝑧=
𝑥 − 𝜇 + 0.5
𝑥 − 𝜇 − 0.5
𝑜𝑟 𝑧 =
𝜎
𝜎
Test 3: Ch 6-7
⦿ For a simple random sample, (1 − 𝛼)100% confidence interval estimator for the population mean 𝜇 is given by
𝑥̅ ± 𝐸 or (𝑥̅ − 𝐸, 𝑥̅ + 𝐸) or 𝑥̅ − 𝐸 < 𝜇 < 𝑥̅ + 𝐸
⦿ The Margin of Error for population mean is given by
𝜎
𝐸 = 𝑧𝛼⁄2 ∙
√𝑛
⦿ For a large sample (𝑛 > 30), (1 − 𝛼)100% confidence interval estimator for the population proportion 𝑝̂ is
𝑝̂ ± 𝐸 or (𝑝̂ − 𝐸, 𝑝̂ + 𝐸) or 𝑝̂ − 𝐸 < 𝑝 < 𝑝̂ + 𝐸
⦿ The Margin of Error for population proportion is given by
𝑝̂ (1 − 𝑝̂ )
𝑛
⦿ The sample statistic for Chi-Square Distribution is given by
(𝑛 − 1) ∙ 𝑠 2
𝜒2 =
𝜎2
⦿ For a simple random sample, (1 − 𝛼)100% confidence interval estimator for the population variance 𝜎 2 is
(𝑛 − 1) ∙ 𝑠 2
(𝑛 − 1) ∙ 𝑠 2
2
<
𝜎
<
𝜒𝑅2
𝜒𝐿2
⦿ For a simple random sample, the (1 − 𝛼)100% confidence interval estimator for the population standard
deviation 𝜎 is given by
𝐸 = 𝑧𝛼⁄2 √
√
(𝑛 − 1) ∙ 𝑠 2
𝜒𝑅2
<𝜎<√
(𝑛 − 1) ∙ 𝑠 2
𝜒𝐿2
⦿ 𝜒𝐿2 is the left-tailed critical value of 𝜒 2 and it is given by
2
𝜒𝐿2 = 𝜒1−𝛼
⁄2
2
 Note that 𝜒1−𝛼⁄2 is a critical value that separates the right area of 1 − 𝛼⁄2.
𝐶ℎ𝑒𝑜𝑛-𝑆𝑖𝑔 𝐿𝑒𝑒
Mathematics Division-Coastal Bend College
e-mail:[email protected]
Page 4
CBC Mathematics Division: Math 1442 Elementary Statistics
Exam Formula Sheets
⦿ 𝜒𝑅2 is the right-tailed critical value of 𝜒 2 and it is given by
𝜒𝑅2 = 𝜒𝛼2⁄2
2
 Note that 𝜒𝛼⁄2 is the critical value that separates the right area of 𝛼⁄2.
Types of Test
If sign used in 𝐻1 is " ≠ "
Left-tailed
Critical Region:
Reject 𝐻0
Critical Value
Two-tailed
Critical Region:
Reject 𝐻0
Critical Region:
Reject 𝐻0
Critical Value
Right-tailed
Critical Region:
Reject 𝐻0
Critical Value
⦿ The 𝑃-value is the probability of getting a value of the test statistic that is as extreme as the one representing the
sample data, assuming that the null hypothesis is true.
 The 𝑃-value for a critical region in Left-tailed tests is the area to the left of the test statistic.
 The 𝑃-value for a critical region in Right-tailed tests is the area to the right of the test statistic.
 The 𝑃-value for a critical region in Two-tailed tests is twice the area in the tail beyond the test statistic.
⦿ The test statistic for proportion is given by
𝑝̂ − 𝑝
𝑧𝑐𝑎𝑙 =
√𝑝(1 − 𝑝)
𝑛
⦿ The test statistic for mean is given by
𝑥̅ − 𝜇𝑥̅
𝑥̅ − 𝜇𝑥̅
𝑧𝑐𝑎𝑙 = 𝜎
or 𝑡𝑐𝑎𝑙 =
𝑠
√𝑛
√𝑛
⦿The test statistic for standard deviation is given by
(𝑛 − 1)𝑠 2
2
𝜒𝑐𝑎𝑙
=
𝜎2
⦿ Decisions and Conclusions
1. 𝑃-value methods:
For the significance level 𝛼,
 If 𝑃-value ≤ 𝛼 for the Left-tailed test, reject the null hypothesis 𝐻0 .
 If 𝑃-value > 𝛼 for the Left-tailed test, fail to reject the null hypothesis 𝐻0 .
 If 𝑃-value ≥ 𝛼 for the Right-tailed test, reject the alternative Hypothesis 𝐻1 .
 If 𝑃-value < 𝛼 for the Right-tailed test, fail to reject the alternative Hypothesis 𝐻1 .
2. Test statistic methods:
 If the test statistic falls within the critical region, reject 𝐻0 or fail to reject 𝐻1 .
 If the test statistic does not fall within the critical region, fail to reject 𝐻0 or reject 𝐻1 .
3. Confidence interval methods:
 If a confidence interval does not include a claimed value of a population parameter, reject the
claim.
 If a confidence interval includes a claimed value of a population parameter, fail to reject the claim.
𝐶ℎ𝑒𝑜𝑛-𝑆𝑖𝑔 𝐿𝑒𝑒
Mathematics Division-Coastal Bend College
e-mail:[email protected]
Page 5