Question

Hi I need help understanding how to do my problems..if I can understand one I can get the rest.
Consider the following hypothesis test:
Ho: U > 20
Ha: U < 20
A sample of 50 provided a sample mean of 19.4 - population standard dev. is 2. Compute the value of the test statistic. What is the p-value and using alpha .05 what is my conclusion?

Answer 1

Ho is spose to be an equality; perhaps H0=20

Ha should represent either what the claim is; or NOT the claim depending on how the claim is worded.

im gonna assume this to be:

Ho=20
Ha<20 ; this gives us a left tailed test since the sign is pointing left.

There tends to be a flow chart on how to determine which test statistic is used, if i recall correctly we have: If the parameter is the mean; and the distribution is normal; and the sample size (n) > 30, and a known "o~" we can use the Z score as out test statistic

the z-score equates to: (given statistic - claim)/(modified sd), in this case I would say:

(19.4 - 20)
Z= ---------- = -.3 sqrt(50) = abt. -2.121...
2 /sqrt(50)

Answer 2

since this is our threshold; anything falling to the left of it will cause us to reject Ho; and anything to the right of it will cause us to "fail to reject" Ho.

The p-value is simply determined by finding the cumulative area (probability) from -inf to -2.121 under the distribution curve; keep in mind that i am assuming this to be a normal distribution.

We can use a z-table to determine the area; look at the column on the left and find 2.1 ; then match that up with the top row that matches .02:


2.12-------------.02
. |
. |
. |
2.1 ------------- (N)
^
this number, in my book at least, gives the area
between the mean and the z-score

In order to use out N value we have to adjust it to find the area NOT between the mean and the z-score.

Lets do this; .5 is all the area up to the mean; so add our N to it

.5 + N = all the area that is NOT what we want so subtract that from 1 (the entire area under the curve equals 1)

1 - (.5+N) is our p-value; to which i get .0170

Answer 3

we compare this to the value of our significance level; that "alpha" value of .05

if our p-value is greater than alpha; we "fail to reject" Ho

if our p-value is less than alpha; we "reject" Ho.

.0170 is LESS THAN .05; so we "REJECT" the null hypothesis, Ho

Answer 4

if you have the ti83 or equivalent; its easier to determine :)

stat--tests--1:(Z-test)-- enter in values

Uo: 20

o~: 2

\(\bar x\): 19.4

n: 50

U: <Uo

Calculate
..............................................

output is:

z = -2.121....
p-value = .01694...

comapre p-value to alpha and determine your conclusion

Answer 5

think of it as the z-score tells us how far we are away from the mean; and the alpha sets a boundary; if out z-score is to far away from the mean it goes out of bounds.

alpha boundary
<.....|...............||.................................|||.................................>
mean
|<----------------------------|
z-score