Topic 3a Continuous Probability Student Notes PDF

3a | Data Analysis & Interpretation - Continuous Probabilities & Normal Distribution Interpret data using statistics tools for business decision-making Intended Learning Outcomes: ❖ Describe the concept of probability in relation to normal distribution. ❖ Discuss characteristics and the related par...

3a | Data Analysis & Interpretation - Continuous Probabilities & Normal Distribution Interpret data using statistics tools for business decision-making Intended Learning Outcomes: ❖ Describe the concept of probability in relation to normal distribution. ❖ Discuss characteristics and the related parameters of continuous probability distribution. ❖ Compute the probability of an event in accordance to the criteria set out by the probability distribution. ❖ Compute the corresponding value in a situation given the probability of occurrence. Odds. Chances. Likelihood. Possibility. These are all words that are often used interchangeably with the word PROBABILITY. From a statistical perspective, what is probability? Probability measures the likelihood of an event happening. It is a value between zero (0) and one (1). It can be expressed in decimals (0.451) or percentages (45.1%). For a situation in which several different outcomes are possible, the probability for any specific outcome is defined as a proportion of all the possible outcomes. Probability of event A = number of outcomes classified by A P(A) total number of possible outcomes Event A collection of one or more outcomes of a study. Outcome A particular result of a study Probabilities that are close to 0 indicate that the chance of the event happening is very unlikely. On the other end of the scale, probabilities that are very close to 1 indicate that the chance of the event happening is very high. BLO1001 Statistics for Business 1|P a g e  Examples of events across the Probability Scale Example Rolling dices are common in many scenarios and activities such as board games. Let’s study the rolling of a standard (fair) six-sided dice. We roll it once. What is the probability that the dice-roll will result in getting an even number appearing face up? We know that the possible outcomes of this study are: Source: Lind, Marchal, & Wathen (2024) We can see there are 3 possible outcomes of an even number, that is rolling a 2, 4 or 6. Hence, Probability of rolling an even number = 3 number of outcomes classified as even numbers = 0.50 P(even number) 6 total number of possible outcomes The probability can also be expressed as a percentage. In this case, there is a 50% probability that you will roll a dice with an even number facing up. Recap You have covered fundamental probability concepts in secondary school and to refresh these concepts, watch these videos before moving on to the next section. This Photo by Unknown Author is licensed Click here: Introduction to Probability Click here: Definition of Intersection, Duration: 04:16min Union, Compliment, Venn Diagram Duration: 4:57min BLO1001 Statistics for Business 2|P a g e Moving on from the dice example, let’s assume we throw the same dice 100 times. If you record the number shown on the top face, the possible outcomes are 1, 2, 3, 4, 5 and 6. Call X the number shown. You record the frequency of appearance for each number and then calculate the probability of each number appearing. The results are tabulated in the table below: We can also use a bar chart to visualise how the probability is distributed across each number. While some numbers may appear more frequently than others (for example, '6' does not appear as frequently as '4'), all numbers have approximately the same frequency of happening, out of the 100 throws. The probability of all possible outcomes must also add up to 1. This brings us to the concept of a probability distribution. Continuous Probability Distribution A continuous probability distribution usually results from measuring a quantitative variable and determining how likely they are to occur. It gives us a picture of the likelihood of all possible outcomes for certain kinds of measurements or data that can change fluidly and are not restricted to set values. There are different types of continuous probability distributions. In our syllabus, our focus is on the normal distribution. BLO1001 Statistics for Business 3|P a g e Focus on normal distribution only Probability in relation to Normal Distribution We learnt about the normal distribution curve in Topic 2a on descriptive statistics for ungrouped data. The normal distribution curve is symmetrical, with the mean, median and mode in the middle and the data points tapering off at the ends of both sides. In topic 2a, we used it to explain the shape of distribution of a set of data. If the shape of distribution is like a normal distribution and the curve is symmetrical then the preferred measure of central tendency is the mean. The normal distribution has the following characteristics: It is a bell-shaped curve and has a single peak at the centre of the distribution. o The measures of central tendency are equal so Mean = Median = Mode and they are located in the centre of the distribution. o The total probability under the distribution = 1.00. Half the area (0.5) under the normal distribution is to the right of the centre line and the other half (0.5), to the left of it. It is symmetrical about the mean. When the normal distribution is cut vertically at the centre value, both sides are mirror images of each other and the area of each half = 0.5. It falls off smoothly in either direction from the central value. The curve gets closer and closer to the X-axis on each side but never actually touches it. The location of a normal distribution is determined by the mean, μ. The dispersion/spread of the distribution is determined by the standard deviation, σ. It indicates how much the data varies from the mean. Extreme values to the very left or very right of the mean are very rare so if you select a data point at random, chances are the value will be quite close to the mean. BLO1001 Statistics for Business 4|P a g e Related Parameters of Normal Distribution Same Mean, Different Standard Deviations Sparkling Logistics is a logistics company that has offices in Australia, Singapore and the USA. In all 3 countries, the average years of service of an employee is 20 years. However, the dispersion of the employees’ years of service are different in each office. How would the normal distribution curve differ for each office? You can see from the left figure, whilst the means are the same across all 3 offices, the different standard deviations caused the distributions to look different. When σ becomes smaller, the distribution becomes narrower, indicating that the years of employee service are closely clustered around the mean. Different Mean, Same Standard Deviation Koko Waffles wanted to understand the distribution of its best-selling cereal as compared to its competitors. It conducted a study and found whilst the mean weights of the cereals for itself and the competitors are different, the standard deviations are the same at 1.6 grams as the figure below. We know the  and  are critical parameters that shape the normal distribution curve. A larger standard deviation results in a wider, flatter curve, signifying greater variability among data points. On the other hand, a smaller standard deviation leads to a narrower, more peaked curve, indicating that the data points are closely clustered around the mean. BLO1001 Statistics for Business 5|P a g e Assume you scored 76 marks for a Marketing test which is a B+ grade. This is good but if you wish to know how well you have performed compared to other students in the cohort, the score of 76 marks will not tell you much. So, more information is needed and your lecturer has shared that the grades are normally distributed. One useful information would be the mean of the cohort. If the mean was µ = 70 marks, you would be in a better position and have performed better compared to another cohort with a µ = 85 marks. So your test result and position relative to the rest of the cohort depends on the mean. But is this accurate and representative enough? Let’s set your score of 76 marks as the x-value. In the above figures, both means are the same at 70 marks. However, the standard deviation,  in (a) is 3 marks while in (b), the  is 12 marks. In (a), when µ = 70 and  = 3, x = 76 is at the extreme right of the normal distribution, indicating that your result is one of the highest in the cohort. In (b), when µ = 70 and  = 12, x = 76 is located between 70 and 82, indicating that your result is slightly above average than the cohort. What does all this imply? Just knowing a value, by itself, does not provide much information. A single value does not tell you whether it is “good” or “bad” relative to everyone else. A single value does not provide information about its position within a distribution. Raw data/values can be difficult to interpret and compare across different distributions. The relative location of your result within the distribution depends on the standard deviation as well as the mean. BLO1001 Statistics for Business 6|P a g e Empirical rule of the normal distribution The empirical rule, often called the "68-95-99.7 rule," is a quick way to remember how data in a normal distribution is spread out around the mean. 1. 68% Rule: About 68% of the data will fall within one standard deviation of the mean. This means that if you measure something like heights of people, around 68% of the heights will be close to the average height, neither too tall nor too short. 2. 95% Rule: About 95% of the data falls within two standard deviations of the mean. So, going further out from the average, most (95%) of the people's heights will be within this range. 3. 99.7% Rule: Finally, about 99.7% of the data will fall within three standard deviations of the mean. This covers almost everyone, including those who are much taller or shorter than average. You may have figured by now it is rather common to see the normal distribution. In fact, it is applicable across many different fields and industries, even in our day-today lives (maybe we just haven’t been noticing ). The Standard Normal Distribution We use the standard normal distribution to find the probabilities for any normal distribution. A standard normal distribution is an important statistical tool to find the probabilities for any normal distribution. Any variable that follows a normal distribution means that it can be "standardised" to be part of a standard normal distribution. ‘Standardising’ a variable is important because it allows us to compare different x-values on the same scale even if they come from different sets of data with different averages or spreads. This “standardisation” involves using the mean and standard deviation to transform each x- value into a z-score/value or standard score. This helps us to identify the exact location of each x-value in a distribution. BLO1001 Statistics for Business 7|P a g e To transform the x-value into a z-score, we use: 𝒙−𝝁 𝒛= 𝝈 where: x = raw or original value of the variable  = mean of the distribution  = standard deviation of the distribution z = location of the x-value in the standard normal distribution z-score specifies the precise location of each x value within a distribution. The sign of the z-score (+ or −) signifies whether the score is above the mean (positive/+) or below the mean (negative/-). The numerical value of the z-score specifies the distance from the mean by counting the number of standard deviations between x and μ. By converting x-values into z-scores, we can see how far away each value is from the average and compare their positions easily. This is especially helpful in understanding probabilities and making meaningful comparisons across different data sets. Shape of Distribution Even after changing the x-values to z-scores, the overall shape of the data (like the curve or distribution) stays the same. This is because we are measuring positions of the x-values differently, in terms of z-scores instead of raw values. When we convert x-values to z-scores, we are essentially ‘rescaling’ the data so that distances are always measured relative to the standard deviation. This helps us compare values consistently, regardless of the original units or scale of the data. Hence, the shape of the data or distribution does not change but our measurement scale is standardised to make comparisons easier. BLO1001 Statistics for Business 8|P a g e Mean in z-scores In a z-score, we measure how far a number (x-value) is from the average (mean) in terms of standard deviations. So, if we ask, "How far is the average from itself?", the answer is always zero since it is exactly at the center of the z-score transformation. Assume a study shows the  to be 100 and when we apply the z-score formula with a x-value of 100 (which also happens to be the  in this case), we get: x−μ 100−100 z= = =0 σ 10 Hence, the standard normal distribution (z-score) will always have a mean of zero (0). Standard Deviation in z-scores The standard deviation in the standard normal distribution follows the empirical rule and is always 1. Negative (- sign) z-scores represent data to the left of the mean (below the mean). Positive (+ sign) z-scores represent data to the right of the mean (above the mean). BLO1001 Statistics for Business 9|P a g e The below figure shows x-values and z-scores on the same distribution. The total area under the curve carries a probability of 1.00. Half the area (probability = 0.50) under the normal curve is to the right of the centre line (the mean) and the other half (probability = 0.50) to the left of the mean. We learnt we need to transform x-values into z-scores so we are able to measure probabilities across different sets of data meaningfully. Before we compute probabilities in different scenarios, we need to know how to use a Standard Normal Distribution Table which lists the probabilities for any normal distribution is used to help us with our computations. Using the Standard Normal Distribution Table (this is based on after having computed the z-score in a question) Assume you have already computed z-score for the question: 1) Round the computed z-score to 2 decimal places and assume you have 0.52. 2) The z-scores are in the left and top margins of the table. 3) To find the probability for a z-score of 0.52, the z-score is split into two parts = 0.5 and 0.02. 4) First, locate 0.5 in the left margin. Next, go to the top and find the column with 0.02. 5) The probability of a z-score value of 0.52 is the intersection of the row of 0.5 and column 0.02, which is 0.1985. BLO1001 Statistics for Business 10 | P a g e Working Example 1 Elena is a Junior Housekeeper with MVP Vacations Group. The amount of tips the housekeeping team receives per shift is normally distributed with a mean of $80 and a standard deviation of $10. a. Some hotel guests are more generous than others in giving tips. What is the probability that Elena receives more than $105 in tips during her shift? Step 1: Define the variable x. Let x = the amount of tips received per shift Step 2: State the mean and standard deviation. Given µ = $80 and  = $10 Step 3: Define the probability statement. We want to determine the probability that Elena receives more than $105 in tips, the probability statement will be: p (x > $105) Steps 4 and 5: Draw a normal distribution with the mean in the centre. At the same time, we will the mark the x-value and shade the appropriate area of interest below the distribution. In this case, we are looking at the area where Elena receives more than $105 in tips. Step 6: Convert the distribution into the Standard Normal Distribution. Transform the x-value into z-score using: 𝐱−𝛍 𝐳= 𝛔 𝟏𝟎𝟓−𝟖𝟎 𝐳> where z = 2.50 𝟏𝟎 The probability statement is now transformed to: 𝟏𝟎𝟓−𝟖𝟎 p (𝐳 > ) 𝟏𝟎 p (z > 2.50) Step 7: Now add the z-axis into the normal distribution and locate the z- score. In this example, z score is +2.50, which is positive and on the right of the mean. +2.50 implies that the x-value is 2.5 standard deviations more than the mean. BLO1001 Statistics for Business 11 | P a g e  Step 8: To find the probability of z-score (2.50), we refer to the Standard Normal Distribution table. When z-score is 2.50, the probability is 0.4938. Recall the characteristics of the normal distribution. It is symmetrical and the total probability under the curve = 1.00. Half the area (0.5) under the normal distribution is to the right of the centre line and the other half (0.5), to the left of it. To find out the probability that Elena receives more than $105 in tips, 𝟏𝟎𝟓−𝟖𝟎 p (x > $105) = p (𝐳 > 𝟏𝟎 ) = p (z > 2.50) = 0.50 – 0.4938 = 0.0062 We conclude with the statement: The probability that Elena receives more than $105 in tips is 0.0062. b. Elena feels that if her total tips for the shift is less than $65, it would be perceived that hotel guests have rated her service level to be poor. What is the probability that hotel guests perceive Elena as providing poor service? 𝟔𝟓−𝟖𝟎 p (x < $65) = p (𝒛 < ) 𝟏𝟎 = p (z < -1.50) = 0.50 - 0.4332 = 0.0668 The probability that hotel guests have perceived that Elena provided poor service is 0.0668. (Note : The z-score reading from the table gives us the probability under the curve from the mean. If we need to find the probability of x < $65 which is the shaded region, we have to take the probability of half the curve (0.5) minus the probability of z-score -1.50) BLO1001 Statistics for Business 12 | P a g e c. What is the probability that Elena receives between $70 and $105 for total tips in her shift? 𝟕𝟎−𝟖𝟎 𝟏𝟎𝟓−𝟖𝟎 p (70 ≤ x ≤ 105) = p( ≤ 𝒛≤ ) 𝟏𝟎 𝟏𝟎 = p ( -1.00 ≤ z ≤ 2.50) = 0.3413 + 0.4938 = 0.8351 The probability that Elena receives between $70 and $105 in total tips is 0.8351. (Reminder : The z-score reading from the table gives us the probability under the curve from the mean. Hence, in this question, we will add up the probabilities of the 2 shaded regions from -1.00 to the mean and then from the mean to 2.50.) Compute the corresponding value in a situation given the probability of occurrence. We have focused on finding probabilities (or area under the normal distribution) based on given 𝑥-values. There are many real-world situations when we are given a scenario with the desired probability and we have to determine the x-value for it. x−μ The same formula (z = ) can be used for the reverse, that is to find the x-value of a given σ probability, when the distribution is normal and µ and  are known. To transform a z-score to an x-value, x−μ z= x = µ + z σ Working Example 2 Mr Tan found that the final marks in his Social Studies cohort are normally distributed with a mean of 72 marks and a standard deviation of 5 marks. He decided to assign his grades for his cohort such that the top 15% of the students will receive an A. What is the lowest mark for a student to achieve an A? Step 1: Define the variable x. Let 𝑥 be the marks of student. Let 𝑥1 be the lowest mark to achieve an A. Step 2: State the mean and standard deviation. µ = 72,  = 5 Step 3: Determine what you need to find. 𝑃 (𝑥 > 𝑥1) = 0.15 BLO1001 Statistics for Business 13 | P a g e  Steps 4 and 5: Draw a picture of the normal distribution with the mean in the centre and shade the appropriate area of interest below the distribution. In this example, the top 15% of students are on the extreme right of the curve. This would mean that the probability is 0.15 which is the area of interest under the curve we are interested in. Step 6: Using the probability, find the nearest z-score from the Standard Normal Distribution table. To find the closest z-score, we apply the empirical rule of a normal distribution. 𝑃 (0