Basic stats help

I need to understand the process for these problems but i have little clue on how to do any of it. >_> The chapter is on hypothesis testing.

 Spoiler

1) Professor Jennings claims that only 35% of the students at Flora college work while attending school. The dean thinks that the professor has underestimated the number of students with part time or full time jobs. A random sample of 81 students shows that 39 have jobs. Do the data indicate that more than 35% of the students have jobs? (use a 5% level of significance)

2) The highway department is testing two types of reflecting paint for concrete bridge end pillars. The two kinds of paint are alike in every respect except that one is orange and the other is yellow. The orange paint is applied to 12 bridges and the yellow to 12 bridges. After 1 year, reflectometer readings were made on all bridge end pillars. (higher reading = higher visibility) For the orange paint, the mean reflectometer reading was X bar1 = 9.4, with standard deviation s1=2.1. For the yellow, the mean was X bar 2 = 6.9, with standard deviation s2 = 2.0. Based on these data, can we conclude that the yellow paint has less visibility after 1 year? (use 1% level of significance)

3) A study is made of residents in pheonix and its suburbs concerning the proportion of residents who subscribe to sporting news. A random sample of 88 urban residents showed that 12 subscribed, and a random sample of 97 suburban residents showed that 18 subscribed. Does this indicate that a higher proportion of suburban residents subscribe to sporting news? Use 5% level of significance.

5) Six sets of identical twins were randomly selected from a population of identical twins. One child was taken at random from each pair to form an experimental group. These children participated in a program designed to promote creative thinking. The other child from each pair was part of the control group that did not participate in the program to promote creative thinking. At the end of the program, a creative problem solving test was given, with the results shown in the following table:

Twin pair | A B C D E F
Experimental group | 53 35 12 25 33 47
Control group | 39 21 5 18 21 42

Higher schores indicate better performance in creative problem solving. Do the data support the claim that the program of the experimental group did promote creative thinking? (use 1% level of significance)

6) The manufacturer of a sports car claims that the fuel injection system lasts 48 months before it needs to be replaced. A consumer group tests this claim by surveying a random sample of 10 owners who had the fuel injection system replaced. The ages of the cars at the time of replacement were (in months): 29, 42, 49, 48, 53, 46, 30, 51, 42, 52.

Thanks a million guys. ;~;

Looks like you should Google level of significance, seeing as how that's in every problem.

(Sorry, never took stats, can't help)

The level of significance is supposed to be what you compare the p-value to at the end to reject or not reject the null hypothesis. If i can get the p-value for each problem then i can solve them all, but getting the p-value is the hard part because finding it depends on the problem...

I solved one of them, so i'll take it out.

I can go through one detailed later if you don't understand, but this is basically what you do

Note: I'm using notation and terminology from my class, so if something is unclear, ask

Ho = Null hypothesis
Ha = Alternative hypothesis

These all sound like LRT (Likelihood ratio tests) tests, also known as Nayman Pearson, and not GLRT (Generalized LRT). Basically you're checking 2 different types of data and comparing them, seeing if they are statistically different with a certain percent of confidence, and if you're checking 1-sided or 2-sided

i.e. You have a coin and you flip it 10 times.

Coin = 7 heads

With 95% confidence (1-alpha, so alpha is 5%) can you say it isn't fair?

Probably with instinct, you can see 7 heads happening, it's not that unlikely and it will still be a fair coin. But say it was 70 heads out of 100, then you will probably say it's unfair.

The main thing you're looking for is the Rejected region (RR), and this is where 2-sded and 1-sided come up in.

Ho = " U = Uo"
Ha = " U !=Uo"

That's a 2 sided test. A 1 sided test would look something like

Ho = " U < or = Uo"
Ha = " U > Uo"

Anyway, I'm assuming you're doing LRT, so you need this formula (I can't really write it out so i'll describe it). Numerator and denominator as follows

On top: The product (Capital Pi symbol) of f ( paramater | x) under Ho

The f(p|x) is the distribution function using the paramter involved with Ho

The bottom: the same thing but it's under Ha (Usually you need to use MLE as your best guess)

Call that quantity lambda. Now you check to see if lambda is less than a constant C, which if it is, you're in the RR, and therefore reject your Ho. C will depend on the level of alpha you chose

If you are not using LRT, you usually just assume that lambda is your best guess, which is X bar. Then you just do this as follows

alpha = P ( |Xbar| < c | Ho) <-- Notice the absolute value around Xbar, this is a sign of a 2-sided test

Now normalize it

alpha = P ( { |Xbar| - Uo } / (sigma / SQRT)n) < {c - Uo} / {sigma / SQRT (n)} )

Now that first quantity you normalized it to the stardard normal Z, so you just have

alpha = P ( Z < c*) where c* is that mess above

Then you get

Z (a/2) = c*

You can then get c*, obtain c, and see if your statistic falls in the RR

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I know that's really confusing, so here are some helpful wiki things to explain it

http://en.wikipedia.org/wiki/Likelihood-ratio_test

Look for: Simple-versus-simple hypotheses

This is when you have

Ho = X
Ha = Y

Where both tests are defined for a single number. When you use LRT, for Ha, just use the value given instead of the MLE like I used in the above example.

After you understand 1 of these (Coins usually easiest), the rest are easy to follow. Same process over and over, just different numbers.