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Hi there, I am in an introduction to statistics class, and I really need help with problem 19.

For problem 19, I think the answer is A because Since this referring this two populations, and the conditions for constructing a confidence interval is satisfied, where the sample is random and independent, large sample, and big population, am I right to think that we are able to construct a 95% confidence interval? I am not sure about how to tie in the proportion of samples being asked in part III.

If you guys could help me with this problem, I would greatly appreciate it. Thanks.

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  • $\begingroup$ Statements I and II only use information derived from the sample of size 27. They are not using the large population! Personally I would not attempt to derive 95% confidence intervals from that small a sample, where each individual is contributing nearly 4% of the infomation. However, as the question is stated, they can be derived (though expect them to be wide :-) Statement III is directly calculable. As the base sample is large we may assume that any sub-sample is distributed as Bin($27,0.59$), so calculate the right tail of $27*0.36$ $\endgroup$ – Scott Burns Dec 9 '16 at 2:23
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Perhaps I am being dim-witted, but every one of those looks like it can be calculated to me.

A different set of issues arises with respect to how to calculate them; how 'crisp' are the confidence intervals; etc.

Technically, a 'statistic' is something/anything you can calculate from the data you have. We are all taught that, if we measure the height of 100 people, the way to estimate the average height is to add them all and divide by 100. We call that process - the add and divide - a 'statistic'.

But a statistic can be anything. For example, you could estimate the average height by measuring only the first person, and ignoring the 99 others. That is also a 'statistic'. Or you could measure every other person, or the last one. You could also use the median (the height of the person who has equally many people taller and shorter) or the mode (the most common height). All of these are 'statistics'.

There are different qualities that one looks for in 'statistics'. For example, being unbiased is one (which, roughly, means that on average it gets you the right answer). Amazingly, just measuring a single person will, with normally distributed heights, be unbiased - as will the average we are all taught to calculate. However, the traditional average will have lower variance, which is why we use it.

Note that many distributions are not symmetrical. So, for example, when we measure average wealth we typically use the median - the person who has as many people richer than he or she as there are poorer. Why? It's the old joke: when Bill Gates walks into a bar, the average person in the bar just got billions of dollars richer. Clearly, the typical average is the wrong measure.

By the way, I hate cute questions like this that seem to show how smart the professor is relative to the students. It really does not help anyone learn anything.

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