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Numerade Educator

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Problem 50 Hard Difficulty

Based on a survey conducted by Greenfield Online, 25 to 34-year-olds spend the most each week on fast food. The average weekly amount of $\$ 44$ was reported in a May 2009 USA Today Snapshot. Assuming that weekly fast food expenditures are normally distributed with a standard deviation of $\$ 14.50,$ what is the probability that a 25- to 34-year-old will spend:
a. less than $\$ 25$ a week on fast food?
b. between $\$ 30$ and $\$ 50$ a week on fast food?
c. more than $\$ 75$ a week on fast food?

Answer

a .0950; b .4933; c .0163

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Video Transcript

okay for this problem were given another normal distribution about the likelihood that people how much money people spend on fast food. So again, with the normal problems, it is good to make a sketch. Uh, what you're trying to find. Um, So the biggest fact here were given is that the mean amount of money spent us $44 and the standard deviation is 14 50 family spend. So is that the general idea like to sketch it out? It's mostly for a showing direction and weaken our steps for this over that right now when we want to find out, I like to sketch it first and then we'll use up an apple. It, in this case to to find the probabilities for this. So for part A, we want to find out how many spent less than $25 to the probability that the X is less than 25 Best show with the diagrams of 40 force here. Another 14. 50 over the side. I'm just gonna give a general sketchy of a relative idea of the size. So 25 is my mark there. Um, and for a part, be sketching. Think we're going to see your 30 to 50. So 30 is going to be roughly right here, and 50 is gonna be over here. So a decent size chunk. So we're gonna find out this area under the curve, which again there in the curve is a probability. Your likelihood that my is spent for that only use the color coding to cash show what we're doing here. But a way to rephrase that teachers probably want you to do that is you say that the probability is between 30 and 50 is written like this 30 50. And finally, for the last part, they want to know it's more than 75 fast food. The probability that the X is greater than $75 per week. We've set 25 to 34 year olds, So let's look at just 75. Before the four was 14. 50 there said I was going to get number over here. You can feel it in If something numbers are viewing more precise, a few teacher expect you to, but it's really more about direction and general idea on. And then I use these scores and scores give us the probability and use table three if you want. I prefer what you've done. A few of these. I like to stop with the one stop shop. Little kind of free apple. It's for normal distributions. You can see they do everything here, but for normal distributions. Ah, click that button. And so the area of normal curve. Well, just like we kind of sketch there. We wanna Karmakar has a center at 44 a standard deviation of 14 50. So you see this and I'm gonna start a gram and I'm gonna take some time. There's a nice cleaner picture that you can actually sketch out the 50 50 and 73 whatnot. So let's find the area first. It's going order under 25 so I don't between two guy, you don't want to the left of a value. Look at 25 you're looking at a table. You'd look at the work 25 is and get the Z score for 25 then see what that area is. Um, the calculation for that area is 250.95 to go back to my read, and it's 0.5 relatively small. Number less than 10% which makes sense. If you look at the size of it. Okay, that probably is going less than $25 per week. Let's do this again, and let's look at our green area. So we want to know between Mary and 50 are changes to between two values. If you're using a table, you take get the two values and then subtract them. But if you use the app, let you say, Well, look between 30 me and 50 messing with the areas you see the area for this problem is about 0.4933 Okay, the likelihood that you spend between 30 and 50 was gonna remark it up. Here 30 and 50 is almost 50% which looks like it's about relative size seems to make sense. And finally, let's look at one to the rights to the right of a value, which is our blue area. We want to know the probability of spending over 75. Now. If you're using a table on this, you get disease for and then be, uh, that probably let it goes bad and you do one minus. That's quirk of the table. When I look left to right. But the apple it's can look to the right of this boundary point, and the area here is 0.163 and there we have it. Just so we have a likelihood of how much money that 25 34 year olds spend free to those. If we're given the median of 44 on the sand aviation, that's the likelihood that you will get for each of those normally distributive points.