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Monthly cell phone bills for a residents of a city are approximately normally distributed and a have a mean of $82 and standard deviation of $11 . Simple random sam...

Question

Monthly cell phone bills for a residents of a city are approximately normally distributed and a have a mean of $82 and standard deviation of $11 . Simple random samples of 100 are drawn and the mean is determined for each sample . what is the probability that the sample mean cell phone is $100 or more ?

Monthly cell phone bills for a residents of a city are approximately normally distributed and a have a mean of $82 and standard deviation of $11 . Simple random samples of 100 are drawn and the mean is determined for each sample . what is the probability that the sample mean cell phone is $100 or more ?



Answers

The monthly utility bills in a city are normally distributed, with a mean of $\$ 100$ and a standard deviation of $\$ 12 .$ Find the probability that a randomly selected utility bill is (a) less than $\$ 70,$ (b) between $\$ 90$ and $\$ 120$ and (c) more than $\$ 140 .$

All right. Now, what is this question? The monthly charges for cellphone plants in the United States are normally distributed. This is very important with mean 62 as and a deviation 18. Okay, so let's just try to draw this. The mean is 62 right? The mean is 62 and standard deviation is 18, which means the points of curvature. The inflection points, right? The point where the graph is going to change its curvature will be 18 apart. Right? What are the inflection? Points? They are new, minus sigma and mu plus sigma. Right. So this will mean that this is a new minus 18. Biggest sigma is given to us as 18. Right? So this is mu minus signal, which is 40 full. All right. And this is Mu plus Sigma, which happens to be 80. These are our points of inflection. Now. The first point was draw the normal curve and label the parameters. Obviously, we have already done that. Then she said the reason that represent the proportion between the plan that charge less than $44 less than 44 is this region, right? This is the region that charges less than $44. Okay, this is Barbie. Now suppose the area under the normal cup to the left of 44 is 440.1587 Now, this area, this shaded reason. Just a moment. Okay, This shaded region is point 158 three 1587 Sorry. 15870.1587 Which means 15.87% of the people 15.87% of people of people get built. That is less than $44. Right? These are monthly cellphone rates. Yeah, and one minus 50.1587 This is around, I think. 8413 Let's just check this one minus 0.1587 Yeah, this is 8413 which was an 84.13% of people get a monthly bill of people, get a monthly bill that is greater than dollar 44. And these are the two interpretations that can be made. Yes. This is how we go about doing this question.

Based on this problem, we know that a cell phone has a lifetime average of being 24.3 months with a standard deviation of 2.6 months. And this is information dealing with the population of cell phones. And in this problem, the company is going to provide 33 employees with the cell phone. So, in essence, we're drawing a sample of cell phones from this population with the 33 cell phones, and the question is asking us what is the probability, or find the probability that they mean, which is the X bar is less than 23.8 months. So because we're talking about a sample of 33 people getting cell phones and we want the distribution of the average of those cell phones were going to apply the central Limit Theorem and the Central Limit Theorem says that the average of the means is the same as the average of the population, which is 24.3, and the standard error of the mean, or the standard deviation of the means is the same as the standard deviation of the population divided by the square root of end. So that's in this case going to be a 2.6 divided by the square root of 33. So we're going to want to draw an image, a bell shaped curve to reflect the problem. And we always put our average in the center and are averages 24.3. And the problem is asking us about being less than 23.8. So we're talking 23.8 right here, and we want to go less than so. The next thing we're gonna need to do is we're going to need to find the Z score associated with 23.8. So the Z score would be 23.8 minus 24.3, divided by standard error of the mean We should be 2.6 over the square root of 33 and the Z score turns out to be approximately negative 1.10 So the 23.8 is negative 1.10 meaning it's 1.10 standard deviations below the mean. So what we can do is we can rewrite this problem instead of talking in terms of the X score we can say that the probability that the averages less than 23.8 is no different than the probability that the Z score is less than negative 1.10 And at that point we would go to the standard normal distribution table in the back of your textbook and the probability that the Z is less than negative 1.10 is 0.13 57 So, to recap, our answer. The question waas that if we give cellphones to 33 people, what is the probability that the average of those 33 cell phone less less than 23.8 months and your answer be 0.1357?

So this question we're told that in one region of the country, the mean amount of credit card debt is 15-15. Standard deviation is 7 1 25 And We're basically as for the probability that the main amount of credit card debt in a sample of 1000 600 households will be Within $300 of the population. Me So that's 15 five, and 15: 50 -300 is 14 950 So for the sample mean we need to send to capture the property of the sample mean allies in these intervals we need the sample Mean, which is 15- 50. Simple. Standard deviation is 71254 square root of 1000 600. So that will be 178 .125. So we have our sample mean and standard deviation. We're basically going to convert, we're going to convert This to our standard normal random variables, we have 15 by 50, so it's within 300. So you know, that's 300 Over 1 7 8.125. And this is negative 300 1 7, 8.125. So that basically gives us between -1.6 ft And 1.68. So that's probably too easy. Less than 1.68-. Probably easy, less than -1.68. So in our table -1.68 is .046 five. So This is 0465 And 1.68. Looking closely at the table is .9535, So .9535 -1465, his point 907

Welcome to new Madrid. In the current problem we are given that the monthly charges for mobile phones are distributed normally. Okay so Arda monthly charges our plans that somebody doing on their phone. So these monthly fees are just is distributed normally. Yes. Me new is equally true, $62 and sigma. Is it one spoke eating Dennis. Okay so the first question that they ask is draw normal distribution. So for that I wouldn't mind getting a standard standard normal distribution image. So I will just select this. Didn't you hear? And Andrew the political trees. So this is a standard normal but we are not having a standard normal distribution, correct? We are considering This distribution. So how will we go if you have to drop the mean is 62 and this each sigma distance is 18. So if you add 6 18 with 62 we get the next one. That is very 600 plus 18. We get zero and each And then again 18 if we add we get 98 that is the next value. Okay? And then the Last one would be six zero. What correct? Because I didn't know. I think I had to put one month six. Yes. So it is 116. And soon we will subtract here it will be four. And for it. So if I add 18 with 4 44 I get 62. Again I have to subtract 18. So six. Yeah And this will be too. So it will be 26. and identify subtract 18. I will have eight Spirit. So this is the diagram. So let me just erase all these unnecessary warnings so that you don't feel confused. The final look of the diagram and I will also rectify this number. Okay? Now the second thing that they want is shared the region that represents the proportion That the charges less than 44. So this is where $44 b. c. Rights of this. This is the bound food. Right? So this is the region there. This is the region where the monthly chart is less than so this is here the charge. Okay Is less than $44. Okay so this is the reason. Now the next question that they ask is supposed the area under the normal gulf to the left of x equals to four people is 1.0.1. That is they're mentioning the value is probability X. Less than 44. Okay. musical through 0.15 87 No they are asking give two interpretations of this result. That means what? Yeah, if you see you were here 44 is equivalent to one sigma LTD. So when we go one sigma limit below the mean. Okay, that means Out of 100% population. Okay. 0.1587 in 200 will give us 15.87%. So if there is a this is 100 then This 15%. Okay, so this is 100 and this is 15%. So this will have monthly charges for uh mobile recharge With less than equal to $44. Okay, so this is the first interpretation. What is it? 15%? of the total population has mm hmm. Monthly plan plans less than $44. Okay. The second interpretation that you can have from this is since we're already using a standard normal curve and you can also see that 44 is nothing. But it's once in movie since $44 is one sigma below the mean. Okay, so since this is so there is. Uh huh. Big share of population who is monthly, mobile monthly phone returns exceeds $44. Because if you think out of 100%, only 15 is going here, remaining 85 is still there, so this remaining 85 for them, the mobile monthly uh reach child is going easily beyond $44. So these are the two interpretations that we can have from this. Okay.


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