STATISTICS AND PROBABILITY.pdf

STATISTICS & PROBABILITY ……………FIRST QUARTER EXAMINATION (GRADE 11) – S.Y. 2024-2025 of the random variable. MODULE 1: RANDOM VARIABLE AND Example: PR...

STATISTICS & PROBABILITY ……………FIRST QUARTER EXAMINATION (GRADE 11) – S.Y. 2024-2025 of the random variable. MODULE 1: RANDOM VARIABLE AND Example: PROBABILITY DISTRIBUTION Suppose two coins are tossed. Let H represent heads, T represent tails and X be the random Random variable is a function that associates a variable representing the number of heads that real number of each element in the sample space. will occur. Find the values of the random variable X. STEPS ON HOW TO DETERMINE THE RANDOM VARIABLES ON ANY EVENTS OR EXPERIMENTS PROPERTIES OF A PROBABILITY DISTRIBUTION 1. The probability of each value of a random Determine the sample space. Assign letters that variable must be between 0 and 1 or will represent each outcome. equal to 0 or 1. We can also write it as 0 ≤ Count the number of the value of the random P(X) ≤ 1. variable (capital letter assigned). 2. The sum of the probabilities of all values of the random variable must be equal to 1 Example: or ΣP(X) = 1. Suppose two coins are tossed. Let H represent heads, T represent tails and X be the random MODULE 4, 5, AND 6: MEAN AND VARIANCE variable representing the number of heads that OF DISCRETE RANDOM VARIABLE will occur. Find the values of the random variable X. Mean is a measure of central tendency that So the possible values of random variable X are 0, balances the distribution. It is also known as 1 and 2. We can also say. average. Mean or average is equal to the sum of the numbers/scores divided by the total number X= 0, 1, 2 of scores. MODULE 2 : EXPLORING DISCRETE & MEAN OF A DISCRETE RANDOM VARIABLE is the CONTINUOUS RANDOM VARIABLES sum of the products of the values of the discrete random variable and their corresponding probabilities. It is also called the Expected Value TWO TYPES OF RANDOM VARIABLES of an event. DISCRETE RANDOM VARIABLE Discrete random variable is a set of possible In symbols: outcomes that are countable or digital. μ = X1 P(X1) + X2 P(X2) + X3 P(X3) + ⋯ + Xn P(Xn) CONTINUOUS RANDOM VARIABLE μ = ∑ [(X) P(X) ] Continuous random variable is a random variable where: where values are on a continuous scale, where μ – is mean or expected value the data can take infinitely X1, X2, X3,..., Xn – are values of the random many values such as temperature, weights and variable heights. P(X1), P(X2), P(X3),... , P(Xn) – are the corresponding probabilities. MODULE 3: CONSTRUCTING PROBABILITY OF TO FIND THE MEAN OF A DISCRETE RANDOM RANDOM VARIABLES VARIABLE Step 1. Construct the probability distribution STEPS IN GETTING THE PROBABILITY OF EACH of the given discrete random variable. VALUE OF THE RANDOM VARIABLES Step 2. Multiply the value of the random Determine the sample space. Assign letters that variable to its corresponding probability. will represent each outcome. Step 3. Find the summation of the products of Count the number of the value of the random the values of the random variable and the variable (capital letter assigned). probabilities. Given the total possible values of the random variable, assign probability values to each value VARIANCE AND STANDARD DEVIATION OF A BySAGOT KITA: REVIEWERS PARA SA LUCIANS BY:(Name) Shawn– Position/Volunteer Ivann P. Balolong - SSLG President 1  STATISTICS & PROBABILITY ……………FIRST QUARTER EXAMINATION (GRADE 11) – S.Y. 2024-2025 DISCRETE RANDOM VARIABLE Step 3. Find the summation of the products of the values of the random variable and the Variance and Standard Deviation are measures probabilities. of dispersion that tell us how spread the scores Example. ∑[(outcome) P(outcome)] = (1/6) + are from the mean and from each other. (2/6) + (3/6) + (4/6) + (5/6) + (6/6) = 21/6 = 3.5 Step 4. Apply the formula in finding the VARIANCE is the average squared deviation from variance: the mean while STANDARD DEVIATION is the σ^2 = ∑[X2 P(X)] − μ2 average deviation from the mean. Step 5. Find the standard deviation by A small variance and standard deviation indicate getting the square root of the variance: that the scores are close to the mean and from σ = √∑[X2 P(X)] − μ2 each other. A large value tells us how spread the scores are MODULE 7: UNDERSTANDING THE NORMAL from the mean. DISTRIBUTION A variance and standard deviation of 0 means that all the scores in the data set are equal. The NORMAL DISTRIBUTION or simply as VARIANCE OF A DISCRETE RANDOM VARIABLE NORMAL CURVE is a bell-shaped distribution Variance of a discrete random variable is the which has an important role in inferential difference between the summation of the statistics. It provides a graphical representation products of the squared value of the random of statistical values variable and its corresponding probability and the square of the mean. PROPERTIES OF NORMAL PROBABILITY DISTRIBUTION In symbols; The distribution curve is bell-shaped. σ^2 = ∑[X2 P(X)] − μ2 The curve is symmetrical about its center. Where: σ^2 – is the variance The mean, median and mode coincide at the X – is the value of the discrete random variable center. P(X) - is the probability of the discrete random The width of the curve is determined by the variable standard deviation of the distribution. μ − is the mean The tails of the curve flatten out indefinitely along the horizontal axis, always approaching STANDARD OF OF A DISCRETE RANDOM the axis but never touching it. That is, the VARIABLE curve is asymptotic to the base line. Standard Deviation of a discrete random The total area under the normal curve is equal variable is the square root of the difference to 1 or 100%. between the summation of the products of the squared value of the random variable and its TWO FACTORS THAT THE GRAPH OF THE corresponding probability and the NORMAL DISTRIBUTION MAY DEPEND ON square of the mean. MEAN In symbols, Mean determines the location of the center of the σ = √∑[X2 P(X)] − μ2 bell-shaped curve. Thus, a change in the value of Where: σ −is the standard deviation the mean shifts the graph of the normal curve to X – is the value of the discrete random variable the right or left. P(X) - is the probability of the discrete random variable STANDARD DEVIATION μ − is the mean Standard deviation determines the shape of the graphs (particularly, the height and width of the STEPS ON HOW TO FIND THE MEAN, VARIANCE, curve). STANDARD DEVIATION OF A RANDOM DISCRETE RANDOM VARIABLE THE STANDARD NORMAL CURVE It is a normal probability distribution that has a Step 1. Construct the probability distribution. mean μ=0 and a standard deviation σ=1 unit. Step 2. Multiply the value of the random Every normal curve (regardless of its mean or variable to its corresponding probability. standard deviation) conforms to the following BySAGOT KITA: REVIEWERS PARA SA LUCIANS BY:(Name) Shawn– Position/Volunteer Ivann P. Balolong - SSLG President 2  STATISTICS & PROBABILITY ……………FIRST QUARTER EXAMINATION (GRADE 11) – S.Y. 2024-2025 “Empirical Rule” also called as 68 – 95 – 99.7. MODULE 9: THE PROBABILITIES AND HOW TO FIND AREAS OF REGIONS UNDER THE PERCENTILES UNDER THE NORMAL CURVE NORMAL CURVE Example 1: Find the area that corresponds to STEPS ON FINDING THE PROBABILITY OF THE each of the following z-score values: NORMAL CURVE a. z = 0.6 Step 1: Draw a normal curve. Step 1: Express the given into two decimal form, Step 2: Locate the z-score value. and it is z = 0.60. Step 3: Draw a line through Step 2: In the z-Table, find the Row z = 0.6 and in Step 4: Shade the region Column with the heading.00. Step 5: Consult the z-Table and find the area that Step 3: The intersection of 0.6 and.00 is 0.7257. corresponds to the value Thus, the area that corresponds to z = 0.60 is 0.7257. MODULE 10: RANDOM SAMPLE PARAMETER/STATISTICS MODULE 8: UNDERSTANDING THE Z-SCORE RANDOM SAMPLING When random sampling is used, each element in Z - SCORE the population has an equal chance of being The z – score is a measure of relative standing selected. Random samples are used to avoid bias that tells how many standard deviations either -error in sampling by selecting the outcome. above or below the mean a particular value is. The scores represent the distances from the PARAMETER center measured in standard deviation units. Parameter (a numerical measure that describes the whole population). IMPORTANCE OF Z - SCORE Raw scores may be composed of large values, Parameters are usually denoted by Greek letters; but these large values cannot be accommodated (μ) population mean at the baseline of the normal curve. So, these (σ2) population variance need to be transformed into scores for (σ) population standard deviation. convenience without sacrificing meanings associated with the raw scores. STATISTICS Statistic (a numerical measure that describes a FORMULAS FOR Z - SCORE sample) Statistics are denoted by Roman letters; The formula in calculating the z –score: (X̅ ) sample mean z = (X−μ)/σ - z-score for the population data (s2) sample variance z = (X−x̄ )/s - z-score for the sample data (s) sample standard deviation. The formula in calculating the raw score X: X = μ + z(σ) MODULE 11: MEAN AND VARIANCE OF THE X = x̅ + z(s) SAMPLING DISTRIBUTION OF THE SAMPLE Note: The raw score X is above the mean if z is MEAN (INFINITE POPULATION) positive and it is below the mean when z is negative. MEAN where: X = the given measurement of a normal The mean of the sampling distribution of the random variable sample means is equal to the mean of the μ = population mean population. Therefore, if a population has a mean σ = population standard deviation μ, then the mean of the sampling distribution of x̅ = sample mean the mean is also μ. s = sample standard deviation μX̅ = μ VARIANCE The variance of the sampling distribution of the BySAGOT KITA: REVIEWERS PARA SA LUCIANS BY:(Name) Shawn– Position/Volunteer Ivann P. Balolong - SSLG President 3  STATISTICS & PROBABILITY ……………FIRST QUARTER EXAMINATION (GRADE 11) – S.Y. 2024-2025 mean is the population variance divided by n, the the normal distribution regardless of the shape of sample size (the number of scores used to the population distribution. The Central Limit compute a mean). Theorem may apply to any sample size but the σ^2X̅ = σ^2/n - for infinite population ideal number of samples is at least 30 so that the data will be more reliable. STANDARD DEVIATION The Standard deviation of the sampling PROPERTY OF SAMPLING DISTRIBUTION OF distribution of the sampling means, which is the SAMPLE MEAN square root of the variance, is also known as the Any sampling distribution of sample mean standard error of the mean. It measures the approaches the normal distribution as the sample degree of accuracy of the sample mean as an size increases. estimate of the population mean. σX̅ =σ /√n - for infinite population μx̅ = μ σ^2x̅ = σ^2 / n MODULE 12: SAMPLING DISTRIBUTION OF THE σx̅ = σ / √n SAMPLE MEAN FOR NORMAL POPULATION COMPUTING THE PROBABILITY OF SAMPLE MEAN WHEN THE VARIANCE IS UNKNOWN WITHIN A GIVEN RANGE z = (X̅ − μ) / (σ / √n) The mean of the sample means (μX) is the same Where X̅ = sample mean as the population mean μ. μ = population mean The variance of the sampling distribution of the σ = population standard deviation sample means is given by: n = sample size σ^2X ̅ = σ^2/n (N−n / N−1) for finite population (consists of a finite or fixed number of elements) MODULE 16: T - DISTRIBUTION The standard deviation of the sampling distribution of the sample means is given By: T - DISTRIBUTION σX̅ = σ/√n · √N−n / N−1 for finite population The t- distribution is similar to a normal where √N−n / N−1 as the finite population distribution which is symmetrical and bell-shape. Correction. The t-table is used when the sample size is small (n < 30) and the population standard deviation σ In general, when the population is large and the is unknown. sample size is small, the correction factor is not The confidence intervals, the degree of freedom used since it will be very close to 1. will always be df = n – 1, or one less the sample size. Computed standard deviation from the population may be different from the standard MODULE 17: THE CONFIDENCE INTERVAL deviation for sampling distribution. This is because of the sampling error. An interval estimate, called a confidence interval, TWO FACTORS THAT INFLUENCE SAMPLING is a range of values or intervals (with lower and ERROR: population variance and sample size upper limits) used to estimate the population parameter. This estimate may or may not contain Sampling error and sample size are inversely the true parameter value. The parameter is related—the smaller the sampling size, the specified as being between two values. It is greater the sampling error. usually in the form of a < Θ < b, which tells that the estimated parameter (Θ) is between two MODULE 13, 14, AND 15: CENTRAL LIMIT values ( a and b ) at a certain level of confidence.) THEOREM POPULATION PROPORTION In the test of population proportions, p stands for CENTRAL LIMIT THEOREM population proportion and p̂ (p-hat) for sample If random samples of size n are drawn from a proportion. Population proportion is a fraction of population, then as n becomes larger, the the population that has certain characteristics. sampling distribution of the mean approaches For example, let’s say you had 1,000 people in the BySAGOT KITA: REVIEWERS PARA SA LUCIANS BY:(Name) Shawn– Position/Volunteer Ivann P. Balolong - SSLG President 4  STATISTICS & PROBABILITY ……………FIRST QUARTER EXAMINATION (GRADE 11) – S.Y. 2024-2025 population and 346 of those people have rapid quarantine pass. The fraction of people who have rapid pass is 346 out of 1,000 or 346/1000. MODULE 18: THE SAMPLE SIZE SAMPLE SIZE The sample size is the number (n) of observations taken from a population through which statistical inferences for the whole population are made. Note: In any case, round up the value obtained to ensure that the sample size will be sufficient to achieve the specified reliability. POPULATION MEAN The minimum sample size needed when estimating the population mean is determined. POPULATION PROPORTION The minimum sample size needed when estimating the population proportion is determined by the formula. BySAGOT KITA: REVIEWERS PARA SA LUCIANS BY:(Name) Shawn– Position/Volunteer Ivann P. Balolong - SSLG President 5