anonymous
  • anonymous
Statistics help!!
Mathematics
jamiebookeater
  • jamiebookeater
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anonymous
  • anonymous
For (a):\[P(\hat{p}\ge0.5)=P\left(\frac{\hat{p}-0.3}{0.023}\ge\frac{0.5-0.3}{0.023}\right)\approx P(Z\ge8.696)=\cdots\]The idea here is that you transform the sample proportion to the \(Z\) statistic using \[Z=\frac{\text{sample mean}-\text{population mean}}{\text{sample standard deviation}}\]
anonymous
  • anonymous
"The proportion p-hat of the sample..." \(\hat{p}\) is the symbol representing the sample proportion. In this case, it's the proportion of students that report a certain opinion. It's important to know that it's the *sample* proportion because it's the statistic you get from the SRS. It may or may not reflect the *population* proportion. Think of it this way: There are billions of people on the planet. We can't ask every person's opinion on some matter, so we narrow our scope. Instead of asking billions of people the same question, we can simplify the task by asking, say, 10 people. We want to make an inference about *everyone* (the population), but we can't, so we use a manageable fraction of everyone (a sample).
anonymous
  • anonymous
The notation \(P(\text{event})\) means the probability of \(\text{event}\) happening, while \(p\) might refer to the population statistic, yes.

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anonymous
  • anonymous
Actually, you're given all the info about the population: "Suppose that \(\textbf{in fact }\bf{30\%}\textbf{ of all students}\) would never answer drugs if asked this question ... In fact, you can assign probabilities to values of p-hat using the normal density curve with \(\textbf{mean }\bf{0.3}\) and \(\textbf{standard deviation }\bf{0.023}\)." What you're doing here is finding the probability that the sample proportion is greater/less than some value. To find these probabilities, you use the population statistics.
anonymous
  • anonymous
To clarify, you're not looking for the actual sample mean - that can only be found by actually collecting data in some experiment or survey. Here, you're just trying to make an inference about what the sample proportion might be based on what you know about the population.
anonymous
  • anonymous
You're not finding \(Z\). You're looking for the probability that this \(Z\) is greater/less than something. You can use a \(z\) table like the one here: http://www.statext.com/tables/Z-Table(GreaterThanZ).jpg This table might not be so useful in this case since there's no probability value associated with numbers greater than \(3.99\). You can notice a pattern here, though. As \(z\) increases, the probability trends toward \(0\). Verifying this with W|A: http://www.wolframalpha.com/input/?i=P%28Z%3E%3D8.696%29 As you can see, the probability is very small, near enough to zero to just say zero.
anonymous
  • anonymous
Yeah. The same process is used to compute the other probabilities.
jim_thompson5910
  • jim_thompson5910
you have the correct z score of z = -2.17391 so far so good
jim_thompson5910
  • jim_thompson5910
do you see how SithsAndGiggles used wolfram alpha to compute the probabilities?
jim_thompson5910
  • jim_thompson5910
you type in P(Z < -2.17391) into wolfram
jim_thompson5910
  • jim_thompson5910
http://www.wolframalpha.com/input/?i=+P%28Z+%3C+-2.17391%29
jim_thompson5910
  • jim_thompson5910
it provides the approximate decimal result along with the drawing (shaded area is very tiny, in blue under the curve)
jim_thompson5910
  • jim_thompson5910
0.015 if you round to 3 decimal places, but yeah
jim_thompson5910
  • jim_thompson5910
ok sounds good
jim_thompson5910
  • jim_thompson5910
what z scores did you get
jim_thompson5910
  • jim_thompson5910
think of it like this raw score of 0.25 means x = 0.25 z = (x-mu)/sigma z = (x-0.3)/0.023 z = (0.25 - 0.3)/0.023 z = ??
jim_thompson5910
  • jim_thompson5910
the "mu" is a lot like the population proportion p the "sigma" is the standard deviation of the sample proportions
jim_thompson5910
  • jim_thompson5910
what z score do you get when you compute z = (0.25 - 0.3)/0.023
jim_thompson5910
  • jim_thompson5910
good
jim_thompson5910
  • jim_thompson5910
and when x = 0.35, what is z?
jim_thompson5910
  • jim_thompson5910
so you'll then type in P( -2.17391 < Z < 2.17391 ) into wolfram alpha
jim_thompson5910
  • jim_thompson5910
The basic steps are this Step 1) Convert all raw scores to standard z-scores Step 2) Use a program, calculator or table to find the area under the curve. A program like wolfram alpha is probably the easiest since you can type in one line and get the answer directly.
jim_thompson5910
  • jim_thompson5910
getting the same
jim_thompson5910
  • jim_thompson5910
so if you randomly picked out a p-hat, there is a 97% chance (roughly) that you'll get a p-hat between 0.25 and 0.35
jim_thompson5910
  • jim_thompson5910
I'll be on for a bit longer, but I cannot help with the test while you're taking it. That's something that needs to be done on your own. I'm sure you'll do fine on the test. I've found that 50% of it is psychological which means that if you're confident, then you'll most likely do well.

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