Statistical Methods Week 6 Sebaran Normal dan Sebaran

Chapter Goals After completing this chapter, you should be able to: § Describe the characteristics of the normal distribution § Translate normal distribution problems into standardized normal distribution problems § Find probabilities using a normal distribution table § Evaluate the normality assumption § Recognize when to apply the uniform and exponential distributions Statistical Methods © 2004 Prentice-Hall, Inc. Week 6 -2 / 82

Chapter Goals (continued) After completing this chapter, you should be able to: § Define the concept of a sampling distribution § Determine the mean and standard deviation _ for the sampling distribution of the sample mean, X § Determine the mean and standard deviation for the sampling distribution of the sample proportion, ps § Describe the Central Limit Theorem and its importance _ § Apply sampling distributions for both X and ps Statistical Methods © 2004 Prentice-Hall, Inc. Week 6 -3 / 82

Probability Distributions Wk. 5 Discrete Probability Distributions Continuous Probability Distributions Binomial Normal Poisson Uniform Hypergeometric Statistical Methods © 2004 Prentice-Hall, Inc. Wk. 6 Exponential Week 6 -4 / 82

Continuous Probability Distributions § A continuous random variable is a variable that can assume any value on a continuum (can assume an uncountable number of values) § § thickness of an item time required to complete a task temperature of a solution height, in inches § These can potentially take on any value, depending only on the ability to measure accurately. Statistical Methods © 2004 Prentice-Hall, Inc. Week 6 -5 / 82

The Normal Distribution Probability Distributions Continuous Probability Distributions Normal Uniform Exponential Statistical Methods ©

The Normal Distribution ‘Bell Shaped’ § Symmetrical § Mean, Median and Mode are Equal Location is determined by the mean, μ § Spread is determined by the standard deviation, σ The random variable has an infinite theoretical range: + to Statistical Methods © 2004 Prentice-Hall, Inc. f(X) σ X μ Mean = Median = Mode Week 6 -7 / 82

Many Normal Distributions By varying the parameters μ and σ, we obtain different normal

The Normal Distribution Shape f(X) Changing μ shifts the distribution left or right. σ

The Normal Probability Density Function § The formula for the normal probability density function is Where e = the mathematical constant approximated by 2. 71828 π = the mathematical constant approximated by 3. 14159 μ = the population mean σ = the population standard deviation X = any value of the continuous variable Statistical Methods © 2004 Prentice-Hall, Inc. Week 6 -10 / 82

The Standardized Normal § Any normal distribution (with any mean and standard deviation combination) can be transformed into the standardized normal distribution (Z) § Need to transform X units into Z units Statistical Methods © 2004 Prentice-Hall, Inc. Week 6 -11 / 82

Translation to the Standardized Normal Distribution § Translate from X to the standardized normal (the “Z” distribution) by subtracting the mean of X and dividing by its standard deviation: Z always has mean = 0 and standard deviation = 1 Statistical Methods © 2004 Prentice-Hall, Inc. Week 6 -12 / 82

The Standardized Normal Probability Density Function § The formula for the standardized normal probability density function is Where e = the mathematical constant approximated by 2. 71828 π = the mathematical constant approximated by 3. 14159 Z = any value of the standardized normal distribution Statistical Methods © 2004 Prentice-Hall, Inc. Week 6 -13 / 82

The Standardized Normal Distribution § § § Also known as the “Z” distribution Mean is 0 Standard Deviation is 1 f(Z) 1 0 Z Values above the mean have positive Z-values, values below the mean have negative Z-values Statistical Methods © 2004 Prentice-Hall, Inc. Week 6 -14 / 82

Example § If X is distributed normally with mean of 100 and standard deviation of 50, the Z value for X = 200 is § This says that X = 200 is two standard deviations (2 increments of 50 units) above the mean of 100. Statistical Methods © 2004 Prentice-Hall, Inc. Week 6 -15 / 82

Comparing X and Z units 100 0 200 2. 0 X Z (μ = 100, σ = 50) (μ = 0, σ = 1) Note that the distribution is the same, only the scale has changed. We can express the problem in original units (X) or in standardized units (Z) Statistical Methods © 2004 Prentice-Hall, Inc. Week 6 -16 / 82

Finding Normal Probabilities Probability is the Probability is measured area under the curve! under the curve f(X) by the area P (a ≤ X ≤ b) = P (a < X < b) (Note that the probability of any individual value is zero) a Statistical Methods © 2004 Prentice-Hall, Inc. b X Week 6 -17 / 82

Probability as Area Under the Curve The total area under the curve is 1. 0, and the curve is symmetric, so half is above the mean, half is below f(X) 0. 5 μ Statistical Methods © 2004 Prentice-Hall, Inc. X Week 6 -18 / 82

Empirical Rules What can we say about the distribution of values around the mean? There are some general rules: f(X) σ μ-1σ μ ± 1σ encloses about 68% of X’s σ μ μ+1σ X 68. 26% Statistical Methods © 2004 Prentice-Hall, Inc. Week 6 -19 / 82

The Empirical Rule (continued) § μ ± 2σ covers about 95% of X’s §

The Standardized Normal Table § The Standardized Normal table in the textbook gives the probability less than a desired value for Z (i. e. , from negative infinity to Z). 9772 Example: P(Z < 2. 00) =. 9772 0 Statistical Methods © 2004 Prentice-Hall, Inc. 2. 00 Z Week 6 -21 / 82

The Standardized Normal Table (continued) The column gives the value of Z to the second decimal point Z The row shows the value of Z to the first decimal point 0. 01 0. 02 … 0. 0 0. 1 . . . 2. 0 P(Z < 2. 00) =. 9772 Statistical Methods © 2004 Prentice-Hall, Inc. 0. 00 . 9772 The value within the table gives the probability from Z = up to the desired Z value Week 6 -22 / 82

General Procedure for Finding Probabilities To find P(a < X < b) when X is distributed normally: § Draw the normal curve for the problem in terms of X § Translate X-values to Z-values § Use the Standardized Normal Table Statistical Methods © 2004 Prentice-Hall, Inc. Week 6 -23 / 82

Finding Normal Probabilities § Suppose X is normal with mean 8. 0 and standard

Finding Normal Probabilities (continued) § Suppose X is normal with mean 8. 0 and standard deviation 5. 0. Find P(X < 8. 6) μ=8 σ = 10 8 8. 6 P(X < 8. 6) Statistical Methods © 2004 Prentice-Hall, Inc. μ=0 σ=1 X 0 0. 12 Z P(Z < 0. 12) Week 6 -25 / 82

Solution: Finding P(Z < 0. 12) Standardized Normal Probability Table (Portion) Z . 00 . 01 P(X < 8. 6) = P(Z < 0. 12) . 02 . 5478 0. 0. 5000. 5040. 5080 0. 1. 5398. 5438. 5478 0. 2. 5793. 5832. 5871 0. 3. 6179. 6217. 6255 Statistical Methods © 2004 Prentice-Hall, Inc. 0. 00 Z 0. 12 Week 6 -26 / 82

Upper Tail Probabilities § Suppose X is normal with mean 8. 0 and standard

Upper Tail Probabilities (continued) § Now Find P(X > 8. 6)… P(X > 8. 6) = P(Z > 0. 12) = 1. 0 - P(Z ≤ 0. 12) = 1. 0 -. 5478 =. 4522. 5478 1. 000 Z 0 0. 12 Statistical Methods © 2004 Prentice-Hall, Inc. 1. 0 -. 5478 =. 4522 Z 0 0. 12 Week 6 -28 / 82

Probability Between Two Values § Suppose X is normal with mean 8. 0 and standard deviation 5. 0. Find P(8 < X < 8. 6) Calculate Z-values: 8 8. 6 X 0 0. 12 Z P(8 < X < 8. 6) = P(0 < Z < 0. 12) Statistical Methods © 2004 Prentice-Hall, Inc. Week 6 -29 / 82

Solution: Finding P(0 < Z < 0. 12) Standardized Normal Probability Table (Portion) Z . 00 . 01 . 02 0. 0. 5000. 5040. 5080 0. 1. 5398. 5438. 5478 P(8 < X < 8. 6) = P(0 < Z < 0. 12) = P(Z < 0. 12) – P(Z ≤ 0) =. 5478 -. 5000 =. 0478 . 5000 0. 2. 5793. 5832. 5871 0. 3. 6179. 6217. 6255 0. 00 Z 0. 12 Statistical Methods © 2004 Prentice-Hall, Inc. Week 6 -30 / 82

Probabilities in the Lower Tail § Suppose X is normal with mean 8. 0

Probabilities in the Lower Tail (continued) Now Find P(7. 4 < X < 8)… P(7. 4 < X < 8) = P(-0. 12 < Z < 0) . 0478 = P(Z < 0) – P(Z ≤ -0. 12) =. 5000 -. 4522 =. 0478 The Normal distribution is symmetric, so this probability is the same as P(0 < Z < 0. 12) Statistical Methods © 2004 Prentice-Hall, Inc. . 4522 7. 4 8. 0 -0. 12 0 X Z Week 6 -32 / 82

Finding the X value for a Known Probability § Steps to find the X value for a known probability: 1. Find the Z value for the known probability 2. Convert to X units using the formula: Statistical Methods © 2004 Prentice-Hall, Inc. Week 6 -33 / 82

Finding the X value for a Known Probability (continued) Example: § Suppose X is normal with mean 8. 0 and standard deviation 5. 0. § Now find the X value so that only 20% of all values are below this X. 2000 ? ? Statistical Methods © 2004 Prentice-Hall, Inc. 8. 0 0 X Z Week 6 -34 / 82

Find the Z value for 20% in the Lower Tail 1. Find the Z value for the known probability Standardized Normal Probability § 20% area in the lower Table (Portion) tail is consistent with a Z -0. 9 … . 03 . 04 . 05 …. 1762. 1736. 1711 -0. 8 …. 2033. 2005. 1977 -0. 7 Z value of -0. 84. 2000 …. 2327. 2296. 2266 ? 8. 0 -0. 84 0 Statistical Methods © 2004 Prentice-Hall, Inc. X Z Week 6 -35 / 82

Finding the X value 2. Convert to X units using the formula: So 20% of the values from a distribution with mean 8. 0 and standard deviation 5. 0 are less than 3. 80 Statistical Methods © 2004 Prentice-Hall, Inc. Week 6 -36 / 82

Assessing Normality § Not all continuous random variables are normally distributed § It is important to evaluate how well the data set is approximated by a normal distribution Statistical Methods © 2004 Prentice-Hall, Inc. Week 6 -37 / 82

Assessing Normality (continued) § Construct charts or graphs § For small- or moderate-sized data sets, do stem-andleaf display and box-and-whisker plot look symmetric? § For large data sets, does the histogram or polygon appear bell-shaped? § Compute descriptive summary measures § Do the mean, median and mode have similar values? § Is the interquartile range approximately 1. 33 σ? § Is the range approximately 6 σ? Statistical Methods © 2004 Prentice-Hall, Inc. Week 6 -38 / 82

Assessing Normality (continued) § Observe the distribution of the data set § Do approximately 2/3 of the observations lie within mean 1 standard deviation? § Do approximately 80% of the observations lie within mean 1. 28 standard deviations? § Do approximately 95% of the observations lie within mean 2 standard deviations? § Evaluate normal probability plot § Is the normal probability plot approximately linear with positive slope? Statistical Methods © 2004 Prentice-Hall, Inc. Week 6 -39 / 82

The Normal Probability Plot § Normal probability plot § Arrange data into ordered array § Find corresponding standardized normal quantile values § Plot the pairs of points with observed data values on the vertical axis and the standardized normal quantile values on the horizontal axis § Evaluate the plot for evidence of linearity Statistical Methods © 2004 Prentice-Hall, Inc. Week 6 -40 / 82

The Normal Probability Plot (continued) A normal probability plot for data from a normal distribution will be approximately linear: X 90 60 30 -2 Statistical Methods © 2004 Prentice-Hall, Inc. -1 0 1 2 Z Week 6 -41 / 82

Normal Probability Plot (continued) Left-Skewed Right-Skewed X 90 60 60 30 30 -2 -1 0 1 2 Z Rectangular Nonlinear plots indicate a deviation from normality X 90 60 30 -2 -1 0 Statistical Methods © 2004 Prentice-Hall, Inc. 1 2 Z Week 6 -42 / 82

The Uniform Distribution Probability Distributions Continuous Probability Distributions Normal Uniform Exponential Statistical Methods ©

The Uniform Distribution § The uniform distribution is a probability distribution that has equal probabilities for all possible outcomes of the random variable § Also called a rectangular distribution Statistical Methods © 2004 Prentice-Hall, Inc. Week 6 -44 / 82

The Uniform Distribution (continued) The Continuous Uniform Distribution: f(X) = where f(X) = value of the density function at any X value a = minimum value of X b = maximum value of X Statistical Methods © 2004 Prentice-Hall, Inc. Week 6 -45 / 82

Properties of the Uniform Distribution § The mean of a uniform distribution is §

Uniform Distribution Example: Uniform probability distribution over the range 2 ≤ X ≤ 6: 1 f(X) = 6 - 2 =. 25 for 2 ≤ X ≤ 6 f(X). 25 2 Statistical Methods © 2004 Prentice-Hall, Inc. 6 X Week 6 -47 / 82

The Exponential Distribution Probability Distributions Continuous Probability Distributions Normal Uniform Exponential Statistical Methods ©

The Exponential Distribution § Used to model the length of time between two occurrences of an event (the time between arrivals) § Examples: § Time between trucks arriving at an unloading dock § Time between transactions at an ATM Machine § Time between phone calls to the main operator Statistical Methods © 2004 Prentice-Hall, Inc. Week 6 -49 / 82

The Exponential Distribution § Defined by a single parameter, its mean λ (lambda) § The probability that an arrival time is less than some specified time X is where e = mathematical constant approximated by 2. 71828 λ = the population mean number of arrivals per unit X = any value of the continuous variable where 0 < X < Statistical Methods © 2004 Prentice-Hall, Inc. Week 6 -50 / 82

Exponential Distribution Example: Customers arrive at the service counter at the rate of 15 per hour. What is the probability that the arrival time between consecutive customers is less than three minutes? § The mean number of arrivals per hour is 15, so λ = 15 § Three minutes is. 05 hours § P(arrival time <. 05) = 1 – e-λX = 1 – e-(15)(. 05) =. 5276 § So there is a 52. 76% probability that the arrival time between successive customers is less than three minutes Statistical Methods © 2004 Prentice-Hall, Inc. Week 6 -51 / 82

Sampling Distributions of the Mean Statistical Methods © 2004 Prentice-Hall, Inc. Sampling Distributions of

Sampling Distributions § A sampling distribution is a distribution of all of the possible values of a statistic for a given size sample selected from a population Statistical Methods © 2004 Prentice-Hall, Inc. Week 6 -53 / 82

Developing a Sampling Distribution § Assume there is a population … § Population size N=4 A B C D § Random variable, X, is age of individuals § Values of X: 18, 20, 22, 24 (years) Statistical Methods © 2004 Prentice-Hall, Inc. Week 6 -54 / 82

Developing a Sampling Distribution (continued) Summary Measures for the Population Distribution: P(x). 3. 2. 1 0 18 20 22 24 A B C D x Uniform Distribution Statistical Methods © 2004 Prentice-Hall, Inc. Week 6 -55 / 82

Developing a Sampling Distribution (continued) Now consider all possible samples of size n=2 16 Sample Means 16 possible samples (sampling with replacement) Statistical Methods © 2004 Prentice-Hall, Inc. Week 6 -56 / 82

Developing a Sampling Distribution (continued) Sampling Distribution of All Sample Means Distribution 16 Sample Means _ P(X). 3. 2. 1 0 18 19 20 21 22 23 (no longer uniform) Statistical Methods © 2004 Prentice-Hall, Inc. 24 _ X Week 6 -57 / 82

Developing a Sampling Distribution (continued) Summary Measures of this Sampling Distribution: Statistical Methods ©

Comparing the Population with its Sampling Distribution Population N=4 Sample Means Distribution n=2 _ P(X). 3 . 2 . 1 0 18 20 22 24 A B C D Statistical Methods © 2004 Prentice-Hall, Inc. X 0 18 19 20 21 22 23 24 _ X Week 6 -59 / 82

Standard Error of the Mean § Different samples of the same size from the same population will yield different sample means § A measure of the variability in the mean from sample to sample is given by the Standard Error of the Mean: § Note that the standard error of the mean decreases as the sample size increases Statistical Methods © 2004 Prentice-Hall, Inc. Week 6 -61 / 82

If the Population is Normal § If a population is normal with mean μ and standard deviation σ, the sampling distribution of is also normally distributed with and (This assumes that sampling is with replacement or sampling is without replacement from an infinite population) Statistical Methods © 2004 Prentice-Hall, Inc. Week 6 -62 / 82

Z-value for Sampling Distribution of the Mean § Z-value for the sampling distribution of where: Statistical Methods © 2004 Prentice-Hall, Inc. : = sample mean = population standard deviation n = sample size Week 6 -63 / 82

Finite Population Correction § Apply the Finite Population Correction if: § the sample is large relative to the population (n is greater than 5% of N) and… § Sampling is without replacement Then Statistical Methods © 2004 Prentice-Hall, Inc. Week 6 -64 / 82

Sampling Distribution Properties Normal Population Distribution § (i. e. is unbiased ) Statistical Methods

Sampling Distribution Properties (continued) § For sampling with replacement: As n increases, decreases Larger sample size Smaller sample size Statistical Methods © 2004 Prentice-Hall, Inc. Week 6 -66 / 82

If the Population is not Normal § We can apply the Central Limit Theorem: § Even if the population is not normal, § …sample means from the population will be approximately normal as long as the sample size is large enough. Properties of the sampling distribution: and Statistical Methods © 2004 Prentice-Hall, Inc. Week 6 -67 / 82

Central Limit Theorem As the sample size gets large enough… Statistical Methods © 2004 Prentice-Hall, Inc. n↑ the sampling distribution becomes almost normal regardless of shape of population Week 6 -68 / 82

If the Population is not Normal (continued) Sampling distribution properties: Population Distribution Central Tendency Variation Sampling Distribution (becomes normal as n increases) Smaller sample size Larger sample size (Sampling with replacement) Statistical Methods © 2004 Prentice-Hall, Inc. Week 6 -69 / 82

How Large is Large Enough? § For most distributions, n > 30 will give a sampling distribution that is nearly normal § For fairly symmetric distributions, n > 15 § For normal population distributions, the sampling distribution of the mean is always normally distributed Statistical Methods © 2004 Prentice-Hall, Inc. Week 6 -70 / 82

Example § Suppose a population has mean μ = 8 and standard deviation σ = 3. Suppose a random sample of size n = 36 is selected. § What is the probability that the sample mean is between 7. 8 and 8. 2? Statistical Methods © 2004 Prentice-Hall, Inc. Week 6 -71 / 82

Example (continued) Solution: § Even if the population is not normally distributed, the central limit theorem can be used (n > 30) § … so the sampling distribution of approximately normal § … with mean is = 8 § …and standard deviation Statistical Methods © 2004 Prentice-Hall, Inc. Week 6 -72 / 82

Example (continued) Solution (continued): Population Distribution ? ? ? Sampling Distribution Standard Normal Distribution Sample ? X Statistical Methods © 2004 Prentice-Hall, Inc. . 1915 +. 1915 Standardize 7. 8 8. 2 -0. 5 Z Week 6 -73 / 82

Sampling Distributions of the Proportion Sampling Distributions of the Mean Statistical Methods © 2004

Population Proportions, p p = the proportion of the population having some characteristic § Sample proportion ( ps ) provides an estimate of p: § 0 ≤ ps ≤ 1 § ps has a binomial distribution (assuming sampling with replacement from a finite population or without replacement from an infinite population) Statistical Methods © 2004 Prentice-Hall, Inc. Week 6 -75 / 82

Sampling Distribution of p § Approximated by a normal distribution if: P( ps). 3. 2. 1 0 § 0 Sampling Distribution . 2 . 4 . 6 8 1 ps where and (where p = population proportion) Statistical Methods © 2004 Prentice-Hall, Inc. Week 6 -76 / 82

Z-Value for Proportions Standardize ps to a Z value with the formula: § If sampling is without replacement and n is greater than 5% of the population size, then must use the finite population correction factor: Statistical Methods © 2004 Prentice-Hall, Inc. Week 6 -77 / 82

Example § If the true proportion of voters who support Proposition A is p =. 4, what is the probability that a sample of size 200 yields a sample proportion between. 40 and. 45? § i. e. : if p =. 4 and n = 200, what is P(. 40 ≤ ps ≤. 45) ? Statistical Methods © 2004 Prentice-Hall, Inc. Week 6 -78 / 82

Example (continued) § Find if p =. 4 and n = 200, what is

Example (continued) § if p =. 4 and n = 200, what is P(. 40 ≤ ps ≤. 45) ? Use standard normal table: P(0 ≤ Z ≤ 1. 44) =. 4251 Standardized Normal Distribution Sampling Distribution . 4251 Standardize . 40 Statistical Methods © 2004 Prentice-Hall, Inc. . 45 ps 0 1. 44 Z Week 6 -80 / 82

Chapter Summary § Presented key continuous distributions § normal, uniform, exponential § Found probabilities using formulas and tables § Recognized when to apply different distributions § Applied distributions to decision problems Statistical Methods © 2004 Prentice-Hall, Inc. Week 6 -81 / 82

Chapter Summary (continued) § Introduced sampling distributions § Described the sampling distribution of the mean § For normal populations § Using the Central Limit Theorem § Described the sampling distribution of a proportion § Calculated probabilities using sampling distributions Statistical Methods © 2004 Prentice-Hall, Inc. Week 6 -82 / 82