Prairie Flower Cereal has annual sales revenue of 400,000,000. George Severn, a 58-year-old sen

step 1 Welcome to numerate. In the current problem, we are given with the probabilities which are for t

Prairie Flower Cereal has annual sales revenue of ...

Key Concepts

Normal Distribution

The normal distribution is a continuous probability distribution that is symmetric about the mean, depicting that data near the mean are more frequent in occurrence than data far from the mean. It is characterized by two parameters: the mean and the variance (or standard deviation). This distribution is a baseline assumption in many statistical models and is especially useful in modeling outcomes that result from the sum of many independent random factors, which makes it a natural choice for approximating production and sales data in many business contexts.

Variance and Standard Deviation

Variance and standard deviation are measures of dispersion in a set of data. Variance quantifies the average squared deviation of each number from the mean, while the standard deviation, being the square root of the variance, provides a measure of spread in the same units as the data. These concepts are crucial in understanding the reliability and consistency of data, such as daily production or sales volumes in a manufacturing or retail setting.

Correlation

Correlation measures the strength and direction of the linear relationship between two random variables. A positive correlation indicates that as one variable increases, the other tends to increase; a negative correlation implies that as one variable increases, the other tends to decrease. In business contexts, understanding the correlation between production and sales can help in forecasting and risk management by indicating how likely these processes are to vary together.

Linear Combinations of Normal Random Variables

When dealing with two or more normally distributed random variables, their linear combination is also normally distributed, provided the variables are jointly normally distributed. This property is extremely useful in financial and operational analysis where outcomes such as profit or cost are expressed as linear combinations (sums, differences, etc.) of individual components like production, sales, and unit costs. The calculation of means and variances of these combinations allows for the assessment of various business scenarios.

Profit Analysis (Revenue and Cost Comparison)

Profit analysis involves comparing total revenue and total costs to assess the financial performance on a given day or over a period. This analysis is crucial for decision making in operations and financial planning. It includes understanding the underlying randomness in revenue and cost data and using statistical methods, such as probability calculations, to estimate the likelihood of profitability under uncertain conditions.

Confidence/Acceptance Intervals

Confidence or acceptance intervals provide a range of values, constructed from observed data, within which a certain parameter (such as the difference between revenue and cost) is expected to lie with a specified probability. This statistical inference tool is used to quantify the uncertainty inherent in sampling processes and to make decisions or accept hypotheses based on the likelihood that the true parameter falls within the computed interval. In business applications, such intervals can guide risk assessments and strategic planning.

Transcript

00:01 Welcome to numerate.

00:02 In the current problem, we are given with the probabilities which are for the values of x and we are also given the px values.

00:17 Okay.

00:18 Now the x is nothing but the weights of weights and pounds.

00:27 So this is 44 pound and all those stuff pounds.

00:31 I will mention.

00:31 Otherwise it will be confusing so 44 45 46 47 48 49 and 50 and then the probabilities are given to be 0 .04 oops and then 0 .13 0 .21 2 .21 0 .29 .0 .0 .9 .0 .0 .9.

01:04 2, 0 .1 and 0 .03.

01:10 Now also we will see that in the next thing that we will be asked is calculate the cumulative probability function.

01:19 So we will calculate that as well directly over here.

01:23 So first we will have that and then we will go towards direct outcomes.

01:28 So 0 .17 because this plus this will be this value.

01:33 Then again this plus this plus this will be this value so this would be 0 .38 now don't think that i'm that fast at calculating i have just simply done the sums and kept it ready because otherwise i will just end up taking way too much time for you and you will keep on waiting while i do my addition so just to try i'm just trying to keep you quicker for the solution so if you see this total this is the checkpoint this is where if you don't get one then just understand there is something wrong with your addition now for the probabilities we will we will start with 44 45 46 47 48 49 and 50 so first things to note these all should be equidistant why are they not because i'm doing it freehand so make sure when you are doing it you do it on a graph paper or use proper scales so that it is equidistant now if you see well i will see the highest probability is this so which is almost 0 .3 so so let's say this is 0 .2 and this is 0 .1 again this also should be equidistant or like at this scales to be maintained now first thing is 0 .04 correct so 0 .04 would be really here okay it's really below next we have 0 .13 which is little about this so we have it here next we have 0 .21 which again is almost at 0 .2 correct so we have it here then 0 .2.

03:40 To 9 correct so we have it here if you see my lines are not accurate but please play along because i'm not able to use any scale or ruler or any other method yes i could have used this always say some drawing objects i believe in in the excel that exists but you know that will end up taking too much of time for us so let's let me try and go with it so 0 .1 if i want to draw that will be something like this and you see it's really like narrow to view and then again i have to go and you know uh take it so the last one it will be 0 .03 so first one was 0 .04 so i have to just reverse draw it so you see this these lines are much more easy.

04:36 To observe while these are not and that's the reason i'm not using them.

04:41 Okay, alternatively we have to now think of drawing this, correct? so yeah, now for drawing this if you see, so first thing what do you observe about this? so this is not a part of question but i'm still adding symmetric almost.

05:04 Well, it is not really symmetric because these values and the this is not.

05:08 This value not seeing but if you see they're almost symmetric so that came out to me when i had drawn the diagram not when i was writing these numbers to be honest so this is how diagrams add values so in any situation you you ever have to convince somebody you can well think of you know keep such an example handy in your mind now again i will start with 44 45 45 4.

05:38 46, 47, 48, 49 and 50.

05:46 So now when i draw this, well, where do i start? now this entire distance has to be 1.

05:55 So i take this highest distance to be 1.

05:57 So this is where my 0 .5 will die.

06:00 So this is where 0 .75 and this is where 0 .25.

06:05 Okay so now if i see 0 .04 so i will be really here okay and then i will go like this and then i will have 0 .17 which is near about 0 .25 but not too much so i go like this and then 0 .38 so again that crosses 0 .25 but not too much near 5 somewhere halfway so i put this and then draw the step and then after this comes 0 .67.

06:37 0 .67 again is near about 0 .75.

06:40 So i again draw a line and go like this.

06:43 And this is called step diagram or cumulative probability diagram.

06:48 So 0 .75.

06:49 And after 0 .75, what do we have? sorry, 0 .67, we have 0 .87.

06:57 So 0 .87 again crosses 0 .75, but not a lot.

07:02 And then we have 0 .97 which is like almost going to touch the one and at 50 it touches one so after 50 it maintains the level okay so this is the cumulative distribution diagram so this is first part this is second part now we go and try to find out the third one and the third one says the third one says what is the probability that a randomly chosen bag will contain more than 45 and less than 49 pounds of mulch, okay? inclusive, correct? so when i say that, that means what? it includes 45.

08:01 That means what? probability more than 45 and less than 49 so think of which region from here we will be taking so if you look at this so we have from 45 to 49 how many weights are there so there are this this this and this right so it is nothing but a sum of all these correct so let me write that this is nothing but probability of 45 plus probability of 46 plus dot dot probability of 49 so that will end up at 0 .13 plus you know 0 .21 and so on i'm sorry i did mark it here i meant 49 only 0 .1 so that would give us 0 .93 again these answers are pre -ready so you don't worry about the calculation now the next part that asks us two packages are chosen randomly and what is the probability that at least one of them okay so probability at least one bag contains okay at least greater than 47 pounds okay so at least one bag will be having this okay so now if you think there are two bags so bag number one bag one okay and bag two now they're asking at least one bag contains the more than 47 thing now so what way we can achieve now think first of all these two events okay so there are a number of bags you choose a bag okay so here is the first bag and you choose another bag so here is the second bag now whatever way will be there these are independent so probability of choosing or the event of choosing the first bag does not impact the second bag selection so therefore probability of choosing one bag with all these conditions okay into probability of choosing second bag with same conditions right and both are basically same correct so what will happen what is the probability of choosing one bag with at least 47 correct so it will be nothing but this this so if you choose this or this or this or this you are satisfying whatever is asked correct so therefore it will be 0 .29 plus 0 .2 plus 0 .1 plus 0 .1 plus 0 .03, correct? again, for the second bag also, same kind of probabilities will happen.

12:06 Correct? so it is again 0 .29 plus 0 .2 plus 0 .1 plus 0 .03, right? so when you add them and multiply them and get the final answer, it should be 0 .3844.

12:22 Now comes a different question that has a small trick in it.

12:29 What is it? so it says that you have to use a computer to find the whatever.

12:36 Right.

12:37 So that's why i have already created this data for you so that you can calculate it.

12:44 So first if you see these are the x values and these are the probabilities.

12:49 Now they're asking how do you find the mean right? how do we find the mean? well for mean if i draw again, so, for mean expectation x, it is nothing but summation of x px, correct? so, i have to multiply x into px, right? so that's what have been done here.

13:15 How have i, how did i do it? i have written a formula is equal to say this is d.

13:23 This column is d.

13:24 And what i am multiplying, i am multiplying a with b.

13:27 So, i have written in the d cell a2 into b2.

13:34 Okay, so let me show you, see, a2 into b2.

13:38 And i just have dragged the same formula.

13:41 So, i have gotten each and every individual item for this.

13:47 Now, summation means what summing all of them? so i have summed it and i have gotten 46 .9.

13:54 Now, see, 46 .9 pound seems...