Question

Statistics help!!

Answer 1

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
\[Z=\frac{\text{sample mean}-\text{population mean}}{\text{sample standard deviation}}\]

Answer 2

"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).

Answer 3

The notation \(P(\text{event})\) means the probability of \(\text{event}\) happening, while \(p\) might refer to the population statistic, yes.

Answer 4

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.

Answer 5

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.

Answer 6

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.

Answer 7

Yeah. The same process is used to compute the other probabilities.

Answer 8

you have the correct z score of z = -2.17391

so far so good

Answer 9

do you see how SithsAndGiggles used wolfram alpha to compute the probabilities?

Answer 10

you type in P(Z < -2.17391) into wolfram

Answer 11

http://www.wolframalpha.com/input/?i=+P%28Z+%3C+-2.17391%29

Answer 12

it provides the approximate decimal result along with the drawing (shaded area is very tiny, in blue under the curve)

Answer 13

0.015 if you round to 3 decimal places, but yeah

Answer 14

ok sounds good

Answer 15

what z scores did you get

Answer 16

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 = ??

Answer 17

the "mu" is a lot like the population proportion p

the "sigma" is the standard deviation of the sample proportions

Answer 18

what z score do you get when you compute z = (0.25 - 0.3)/0.023

Answer 19

good

Answer 20

and when x = 0.35, what is z?

Answer 21

so you'll then type in P( -2.17391 < Z < 2.17391 ) into wolfram alpha

Answer 22

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.

Answer 23

getting the same

Answer 24

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

Answer 25

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.