Use the given information to find the​ P-value and the critical​ value(s).

Question

Assume that the significance level is

alpha equals 0.05 α=0.05.

Use the given information to find the​ P-value and the critical​ value(s).

The test statistic of

z=1.58

1.58 is obtained when testing the claim that

p greater than 0.7 p>0.7.

Answer 1

Alexis,

A hypothesis test can be calculated in several ways (test whether sample mean is within critical interval around the hypothesized mean, compare the Z score with a critical Z value, or compare the test p value with alpha). All will lead to the same conclusion.

Your null hypothesis (nothing has changed - we maintain status quo) is H(0): p = 0.7

Your alternative hypothesis (something is going on) is H(a): p>0.7

Your test is to see whether or not the sample you took reveals something different about what you assumed initially.

1) The critical value for Z is calculated directly from the significance level:

For a one-tail test, Z crit = Z(1-alpha) = 1.64; if Z score > Z critical, we reject H(0)

The Z score for the test should land further away than 1.64 in order to reject the null.

Z is simply a number of standard deviations (standardized measurement of distance away from the mean); the further away you are from the mean, the less likely it should have happened.

2) The critical values for p (lower limit and upper limit) are calculated from alpha, p and n (sample size)

LL = p - Z crit * √(pq/n)

UL = p + Z crit * √(pq/n)

Since we do not have a sample size, nor do we have a p-hat for the sample mean, we cannot calculate the critical interval for p... skip...

3) the p value can be calculated from the Z score (either from a Z table or in Excel):

p = 1-NORM.S.DIST(1.58,1) = 0.057

if p < alpha, we reject the null

the p value represents the probability the event (sample measured) happened by chance alone. If this probability is low, we cry victory and "claim" we found something different that what we originally hypothesized.

In both cases (1 and 3), we cannot reject the null, since Z score < Z crit and p > alpha. Given how close p is from alpha and Z from Z crit, it remains a close call (a slightly higher significance level of 0.06 would have changed your claim.

If you ever find yourself where one solution seems to reject the null, and the other does not, check your math again!

As you see the Z crit is calculated ONLY from the significance level, whereas the p value is calculated ONLY from the Z score. The Z critical defines the fence (where you should land), the Z score defines where you actually landed. Alpha is the probability of rejecting the claim (arbitrary) and defines the risk you are willing to take to make a mistake; the p value is the probability (chance) of landing where you did.

alpha = 0.05 means you accept the risk of making an error 5% of the time; one sample out of 20 will land in the rejection region (by chance alone). Claims made with loosely chosen significance levels are risky...

Hope this helps