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8) A random sample of 45 life insurance policy holders showed that the average premiums paid on their life insurance policies was S342 per year with a standard devi...

Question

8) A random sample of 45 life insurance policy holders showed that the average premiums paid on their life insurance policies was S342 per year with a standard deviation of S58. Construct a 90% confidence interval for the population mean.

8) A random sample of 45 life insurance policy holders showed that the average premiums paid on their life insurance policies was S342 per year with a standard deviation of S58. Construct a 90% confidence interval for the population mean.



Answers

Construct the appropriate confidence interval. A simple random sample of size $n=12$ is drawn from a population that is normally distributed. The sample mean is found to be $\bar{x}=45$ and the sample standard deviation is found to be $s=14 .$ Construct a $90 \%$ confidence interval about the population mean.

Given the following information for two independent randomly selected samples and were asked to provide and 90% confidence interval for the difference between two means now a 90% confidence interval where the difference between two means is written as or estimated as the difference between the sample means plus or minus the critical value. Times the square root of sample variance for a sample, one over the size of sample one plus the sample variance, for example to over the size of sample too. So that's that's our confidence interval and as you may have noticed, we already have sample averages. We have the sample variances and we have the sample sizes. So the only parameter left to find is the critical value, and the first step is to determine the degrees of freedom when you're calculating manually, as we're doing now, degrees of freedoms estimated by the smaller of N one minus one or an to minus one. So this time and two is 15 so that will be the smaller value so and to minus one is equal to 14 so the critical value. So in her case, Alfa, the confidence interval is 90% confidence interval. So Alfa is one minus 0.9, which is 0.10 And if we go to a tea table to look this up, degrees of freedom is 14. Its significance level is 10%. So we're looking for 0.10 area in the two tails and that gives us 1.761 So now we can just plug these numbers into our confidence, interval the equation. So we have 35 minus 30 plus or minus 1.761 and this is equal to you five plus or minus 6.42 Which tells us that the confidence interval ranges from minus 6.3 to 16.3.

In this exercise, we're going to be interpreting The 90% confidence interval for the population. Mean difference me one minus mewtwo Being given to range from -10 to -5. No, this means we can be 90% uh confident That me one when this mewtwo lies somewhere between -10 and -5. In other words, we could say that we are 90% confident that new one is somewhere between five and 10. Less than me too, so NEW one is less than new two by how much something between five and 10.

In this exercise, we're going to be interpreting the 90% confidence interval for the population mean difference. New One menace mute. Um If you were given that it ranges from 5 to 10. Now, the interpretation is as follows that we can be 90% confident that new one minus meWTwo lies somewhere between five and 10. In other words, We can say that we are 90% confident that new one is somewhere between five and 10 greater than new too, so new one is greater than Uh Mewtwo by how much something between five And 10. So new one is greater than YouTube by how much something between five and 10.

Following a solution for number 16 and this study's theme park spending. So there was a random sample of 40 visitors and the mean amount spent per person per day at that particular theme park is $93.43. That's a lot. And it says the population standard deviation is 15, so we actually have sigma here, the population standard deviation sigma is 15 and we're asked to find the 95% confidence interval for the mean amount spent per person per day at this particular um theme park. So what we're gonna do here is determine which method to use and it's going to be the Z interval. The reason why it's a Z interval. Well one, we're testing, not testing, we're estimating the mean, so it's got to be there the tear the Z interval and were able to use the Z interval because we actually know what sigma is. So um we know the population standard deviation. We can be a little bit more accurate with that Z interval. So you can use the z interval formula or you can use technology. I'm gonna use technology because it goes by a little bit more quickly. But if you go to stat and then test, it's the seventh option here is the interval And just make sure summary stats is highlighted there. And remember sigma, the population standard deviation was 15, The x bar was 93.43 And the sample size was 40 visitors and we want to be 95% confidence of 95 is the sea level. And then whenever we calculate this top band here, that gives us the confidence interval. So 88.78 and 98.08, I'm rounding to the nearest sent here, so I'm rounded to two decimal places. So 88 point 78 and 98 08 So a little dollar signs there. Okay, so that's my confidence interval now, I actually have to interpret that confidence interval with um with the theme at hand. So here's how these confidence intervals, they kind of follow the same pattern. But we're gonna say We are 95% confident I'm typing it because it'll be neater this way that the mean amount of money all people spend daily per person at this theme park Is between 88. Just put dollar in front of it, 88 points mm 0.78 and $98.08. Okay, so we're 95% confident that the mean amount of money that all people spend daily per person at this theme park is between $88.78.98.08.


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