Population

Question

 The general population (Population 2) has a mean of 30 and a standard deviation of 5, and the cutoff Z score for significance in a study involving one participant is 1.96. If the raw score obtained by the participant is 45, what decisions should be made about the null and research hypotheses? Report the z-score and your decision. (show all work)

Answer 1

Theresa:

 

I thought someone would answer my "salvaged" question.  Well, I will do that too.  Please read it from my comment.

 

Step 1: Whether random sample, whether variable is on interval-ratio scale, whether n is large do not apply here, we are given only one observation, FROM a normal  ppulation.

Step 2:Null and alternative (research) hypotheses, alpha.

 

We are given the critical value of z as 1.96.  We are asked "For what significance level does this value, 1.96, correspond?"  This value can correspond to either a one-sided or a two-sided test.  Let us choose the more common two-sided test.  This means we have to find α such that P|Z| > 1.96] = α.  From standard normal tables, alpha = 0.05.  Incidentally, if the one sided test rejects when computed z_score is greater than 1.96, P[ z_score > 1.96] = 0.025, the level of sig for a one-sided test.

 

We are given that the population mean is 30 and the SD is 5. For a two sided test that rejects when P|Z| > 1.96] = α,  the null and the alternative hypotheses are:

 

H_o : μ = 30,  H₁:  μ≠ 30.

 

Step 3. Test Statistic and its sampling distribution.  We have only 1 observation, so the statistic, the sample mean, is itself.  Its sampling distribution is the same as that of the original population, normal with mean 30 and SD 5. The test statistic is the z_score = (X- mean)/ SD.  Xis observed to be 45.

 

Step 4.  Compute the test statistic,

 

z_score = (45-30)/5 = 3.

 

Step 5: The P_Value, being the probability of obtaining the calculated value, or worse (favoring H₁) is obtained as twice P(Z>3) = 2(0.00135) = 0.0027, very small.  Hence based on the P_value our data has a very strong evidence against the null hypothesis.  Based on the critical value, since 3 is greater than 1.96, the data favors rejection of the null hypothesis.

 

Other than the word "participant" there is no context here in the question, so we are done.

 

 

Dr. G.