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Exam Formula Sheets
๐ถโ๐๐๐-๐๐๐ ๐ฟ๐๐
Test 1: Ch 1-
โฆฟ Relative Frequency(RF) =
Class Frequency Sum of All Frequencies
โฆฟ Percentage Frequency = RF ร 100%
โฆฟ Angle Frequency = RF ร 360ยฐ
โฆฟ Class Boundaries = Upper limit of one class + Lower limit of next class 2
โฆฟ Class width =
Maximum value โ mininum value Number of classes
โฆฟ Class Midpoint =
Lower Class Limit + Upper Class Limit 2
โฆฟ Population mean ๐ =
๐
๐ผ=
โฆฟ Sample mean ๐ฅฬ
=
๐
๐=
โฆฟ Midrange = Largest value + Smallest value 2 โฆฟ Range = (Maximum value) โ (Minimum value)
Skewed to the left (Negative skewed)
Normal (Bell-shaped)
Skewed to the right (Positve skewed)
Mean โค Median โค Mode Mean = Median = Mode Mode โค Median โค Mean
Arrange the data in ascending order (Smaller to bigger number)
Is the number of data (๐) Odd or Even
Even
Finding Median
The average of the values of ๐ 2
๐กโ and ๐ 2 + 1
๐กโ position
Odd The value at ๐+ 21
๐กโ position
Exam Formula Sheets
๐ถโ๐๐๐-๐๐๐ ๐ฟ๐๐
โฆฟ Mean Absolute Deviation (MAD) =
๐
๐=
โฆฟ Population Variance ๐^2 =
โ(๐๐ผ โ ๐)^2
๐
๐ผ=
, where ๐ =
๐
๐ผ=
โฆฟ Population Standard Deviation ๐ = โ๐^2
โฆฟ Sample Variance ๐ ^2 =
โ(๐ฅ๐ โ ๐ฅฬ
)^2
๐
๐=
or
โ ๐ฅ๐^2 โ (โ ๐ฅ๐)
2 ๐ ๐ โ 1 or
๐(โ ๐ฅ๐^2 ) โ (โ ๐ฅ๐ )^2
โฆฟ Sample Standard Deviation s = โ๐ ^2
โฆฟ Coefficient of variation = CV =
โ 100%, for a sample.
โฆฟ Coefficient of variation = CV =
โ 100%, for a population.
โฆฟ The empirical rule for normal distribution is defined as
- 68% of data values fall in 1 standard deviation (๐)^ of the mean. = 68% of data values fall in the interval (๐ โ ๐, ๐ + ๐)
- 95% of data values fall in 2 standard deviation (2๐) of the mean. = 95% of data values fall in the interval (๐ โ 2๐, ๐ + 2๐)
- 99.7% of data values fall in 3 standard deviation (3๐)^ of the mean. = 99.7% of data values fall in the interval (๐ โ 3๐, ๐ + 3๐)
โฆฟ ๐ โ 2๐ โค Usual values โค ๐ + 2๐
โฆฟ Unusual values < ๐ โ 2๐ or Unusual values > ๐ + 2๐
โฆฟ Chebyshevโs theorem: 1 โ
๐^2
โฆฟ ๐งscore: ๐ง =
โฆฟ Complement of event ๐ธ is denoted by ๐ธ๐^ or ๐ธฬ
: ๐(๐ธ) + ๐(๐ธฬ
) = 1
โฆฟ Multiplication Rule
๐(๐ด โฉ ๐ต) = ๐(๐ด) โ ๐(๐ต|๐ด)^ for Dependent Events ๐(๐ด โฉ ๐ต) = ๐(๐ด) โ ๐(๐ต) for Independent Events
๐ โ 3 ๐ ๐ โ 2 ๐ ๐ โ ๐ ๐ ๐ + ๐ ๐^ +^2 ๐๐ + 3 ๐
Exam Formula Sheets
๐ถโ๐๐๐-๐๐๐ ๐ฟ๐๐
โฆฟ Poisson Distribution:
๐(๐ฅ) =
โฆฟ ๐งscore: ๐ง =
โฆฟ Central Limit Theorem
- Mean of all values of ๐ฅฬ
= ๐๐ฅฬ
= ๐
- Standard deviation of all values of ๐ฅฬ
= ๐๐ฅฬ
=
โฆฟ Normal Distribution to Approximate Binomial Probabilities
If ๐๐ โฅ 5 and ๐๐ โฅ 5 for a Binomial Probability, then
๐ = ๐๐; ๐ = โ๐๐๐; ๐ง =
Test 3: Ch 6-
โฆฟ For a simple random sample, (1 โ ๐ผ)100% confidence interval estimator for the population mean ๐ is given by ๐ฅฬ
ยฑ ๐ธ or (๐ฅฬ
โ ๐ธ, ๐ฅฬ
+ ๐ธ) or ๐ฅฬ
โ ๐ธ < ๐ < ๐ฅฬ
+ ๐ธ โฆฟ The Margin of Error for population mean is given by
๐ธ = ๐ง๐ผ 2โ โ
โฆฟ For a large sample (๐ > 30), (1 โ ๐ผ)100% confidence interval estimator for the population proportion ๐ฬ is ๐ฬ ยฑ ๐ธ or (๐ฬ โ ๐ธ, ๐ฬ + ๐ธ) or ๐ฬ โ ๐ธ < ๐ < ๐ฬ + ๐ธ โฆฟ The Margin of Error for population proportion is given by
๐ธ = ๐ง๐ผ 2โ โ
โฆฟ The sample statistic for Chi-Square Distribution is given by
๐^2 =
(๐ โ 1) โ ๐ ^2
๐^2
โฆฟ For a simple random sample, (1 โ ๐ผ)100% confidence interval estimator for the population variance ๐^2 is (๐ โ 1) โ ๐ ^2 ๐๐
^2
< ๐^2 <
(๐ โ 1) โ ๐ ^2
๐๐ฟ^2
โฆฟ For a simple random sample, the (1 โ ๐ผ)100% confidence interval estimator for the population standard deviation ๐ is given by
โ
(๐ โ 1) โ ๐ ^2
๐๐
^2
(๐ โ 1) โ ๐ ^2
๐๐ฟ^2
โฆฟ ๐๐ฟ^2 is the left-tailed critical value of ๐^2 and it is given by ๐๐ฟ^2 = ๐1โ๐ผ 2^2 โ ๏ Note that ๐1โ๐ผ 2^2 โ is a critical value that separates the right area of 1 โ ๐ผ 2โ.
No
Normally distributed?
๐ > 30 Other Methods
Population
No
Exam Formula Sheets
๐ถโ๐๐๐-๐๐๐ ๐ฟ๐๐
โฆฟ ๐๐
^2 is the right-tailed critical value of ๐^2 and it is given by ๐๐
^2 = ๐๐ผ 2^2 โ ๏ Note that ๐๐ผ 2^2 โ^ is the critical value that separates the right area of ๐ผ 2โ^.
โฆฟ The ๐-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 ๐-value for a critical region in Left-tailed tests is the area to the left of the test statistic. ๏ The ๐-value for a critical region in Right-tailed tests is the area to the right of the test statistic. ๏ The ๐-value for a critical region in Two-tailed tests is twice the area in the tail beyond the test statistic. โฆฟ The test statistic for proportion is given by
๐ง๐๐๐ =
โฆฟ The test statistic for mean is given by
๐ง๐๐๐ =
or ๐ก๐๐๐ =
โฆฟThe test statistic for standard deviation is given by
๐๐๐๐^2 =
(๐ โ 1)๐ ^2
๐^2
โฆฟ Decisions and Conclusions
- ๐-value methods: For the significance level ๐ผ, ๏ If ๐-value โค ๐ผ for the Left-tailed test, reject the null hypothesis ๐ป 0. ๏ If ๐-value > ๐ผ for the Left-tailed test, fail to reject the null hypothesis ๐ป 0. ๏ If ๐-value โฅ ๐ผ for the Right-tailed test, reject the alternative Hypothesis ๐ป 1. ๏ If ๐-value < ๐ผ for the Right-tailed test, fail to reject the alternative Hypothesis ๐ป 1.
- Test statistic methods: ๏ If the test statistic falls within the critical region, reject ๐ป 0 or fail to reject ๐ป 1. ๏ If the test statistic does not fall within the critical region, fail to reject ๐ป 0 or reject ๐ป 1.
- 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.
Types of Test
Left-tailed (^) Two-tailed Right-tailed
If sign used in ๐ป 1 is " โ "
Critical Region: Reject ๐ป 0
Critical Value Critical Value Critical Value
Critical Region: Reject ๐ป 0
Critical Region: Reject ๐ป 0
Critical Region: Reject ๐ป 0
