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A marketing research company carried out a survey to determine the mean the mounds that smokers spend on cigarettes. The company found that the distribution of the ...

Question

A marketing research company carried out a survey to determine the mean the mounds that smokers spend on cigarettes. The company found that the distribution of the amount spent by week tended to follow a normal distribution, with a standard deviation of $5. a sample of 49 smokers revealed that an average of $20. with 95% confidence level, compute the confidence interval for the mean amount that smokers spend on cigarettes in one week.

A marketing research company carried out a survey to determine the mean the mounds that smokers spend on cigarettes. The company found that the distribution of the amount spent by week tended to follow a normal distribution, with a standard deviation of $5. a sample of 49 smokers revealed that an average of $20. with 95% confidence level, compute the confidence interval for the mean amount that smokers spend on cigarettes in one week.



Answers

In a random sample of 40 visitors to a certain theme park, it was determined that the mean amount of money spent per person at the park (including ticket price) was $\$ 93.43$ per day. Assuming the population standard deviation of the amount spent per person is $\$ 15,$ construct and interpret a $95 \%$ confidence interval for the mean amount spent daily per person at the theme park.

Following a solution to number one and number 11, and this looks at aggravated assault where they randomly sample 40 felons, um and they look at the average Sentencing for those 40 felons and that it's 54 months is the average sentence that they all have with a standard deviation of eight. Now it doesn't actually say population standard deviation, it just says sample standard deviation or the standard deviation of the samples eight months, And that's the length of sentencing, and we're asked to find the 95% confidence interval and we need for the for the population mean sentencing and we're going to use the tea interval. Um The reason why we use a T interval is because we're estimating population mean, you in this case means sentence time and we don't know what the population standard deviation is, We only know what sigma is. I'm sorry, we only know what s is the sample standard deviation? So we have to use the tea interval so you can either use the formula or any sort of technology one, I'm gonna use the T. I. T. Four because it works out pretty nice nicely. So if you go to stat on the T. Four and air over two tests and just go to this eighth option here, the T interval and make sure summary stats is highlighted there and the X bar remember was 54 months and that standard deviation was eight and the sample size was 40 felons and we want to be 95% confidence, we're going to change this 2.95 the sea level. And then whenever we calculate this top band here, that gives us the confidence interval. So let's go and write that down. So 51 point 441 and 56 559 So it doesn't say to do it yet, but if we were to interpret this interval, we would just say we are 95% confident that the mean sentencing time for all convicted felons with aggravated assault Is between 51.441 and 56.559 months.

We are very familiar with the methods. By now we want to construct the confidence level of 99 point Sorry off 99% and we have 20 different readings. So in order to control the confidence interval, what I've done is I've again simply input the values over here. These you can see we have 20 values. We are example mean a study 2.5 and a standard deviation is 20.3, you know, confidence level of 99. I select 99 this is the confidence interval that I get 20.8 to 44.2 20.8 20 pointed 20.8 to 44 point to This is my 99% confidence in trouble. Alright. Could also have done this by using the same old formula here and would have become 20 degree of freedom would be 19. Substitute these values Alfa. I would have become 0.50 point 00 fight and put the values in this formula and we come to the same result. This one now it is asked. Does this say? What is the question? There's the conference interval. Give us a good information about the population of all cancers off the same pretty brands that I consumed. No, because we have taken only one can off every brand in this, and only one can cannot be a representative off all the cans off that particular brand. So the answer is no. And we don't even care if this is normally distributed, because this is simply not going to give us a good idea off the, you know, off the mean off the mean off the mean of what we are. Yeah, the mean off caffeine for 12 rounds off a drink. So we don't even care if this is normally distributed.

Now, this is the question that we have in this case. What is our end in his 20? So degree of freedom is going to become 19 inches 20 So D efs 19. This is the formula that I use for my standard deviation. Well, for by two. Since I'm constructing a 95% confidence interval, this is going to be 0.25 and I can again put in the values over here. I can find the value of the alphabet to using a table, substitute the values and I can get my confidence in trouble. Or I can simply use a software as I have done. Nobody care. Absolutely sorry. I have just simply put in the values. And this calculator has calculated all of the world is for me. You can see the sample mean the sample standard deviation on the confidence level of selected as 95. They can select whatever we want. A 2nd 95 and the conference of Devil s 4.96 to 8.44 point 964.96 to 8.4 This is my confidence interval now. Does it appear. The college students typically earned a bachelor degrees in four years. In four years. No, you can see that this confidence interval does not contain four years. So we can say that on an average means it is at 95% confidence. We can say that it takes more than 4.96 years to get a bachelor's degree. Is there anything about the data that would suggest that the confidence interval might not be a good result? Well, first of all, this is a very small sample size. This is a very small sample size, and the second thing is, uh, college Soon bachelor's degree based on data from Yeah, I think over here, the only problem is that the sample size is very small, so we cannot generalize it for the entire population.

The following is a solution # six. And this asks 785 people whether they follow college football. And of those 785 people, 275 of them said that they followed college football. So that was the prompt. And we're asked to find a 95% confidence interval in this scenario. So if you wanted to you could find the P. Hat. You just take the X. Over the end but you really don't need to. Especially if you're gonna use your calculus which I'm going to show you here in a second. But to 75 out of 7 85 um That's gonna be our P. Hat. And uh this is a proportion uh they in fact it's a one proportion Z interval. Okay? Because it doesn't ask anything about mean or anything like that. It just gives you a proportion. So that's what we're gonna do. So you can use the formula if you so wish it just may take you a little longer. I'm gonna go to the T. I. T. Four and if you go to stat and air over two tests and you go all the way down to the a option where it says one prop Z ent we're doing a one proportions E interval. And uh the X value remember was 2 75. That was the number of favorable the people that said that they follow college football out of a total possible 785 people that were randomly selected. And we want to be 95% confidence of 950.95 will be the sea level. And then whenever we calculate that this top band here that's our confidence interval. Now you can write the P. Hat if you want. It's about 35% but this top band is actually the answer. So I'll go in round I liked around three decimal places, so .317 or 31.7 And .384 or 38.4%. Now it doesn't actually say to do this. But if you wanted to interpret this, you would just say we can be Um, confident that the true population proportion of people who follow college football is between 31.7 and 38.4%.


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