A sample of 42 observations is selected from one population with a population st

Question

A sample of 42 observations is selected from one population with a population standard deviation of 3.1. The sample mean is 101.0. A sample of 52 observations is selected from a second population with a population standard deviation of 4.0. The sample mean is 99.8. Conduct the following test of hypothesis using the 0.04 significance level.

State the decision rule. (Negative amounts should be indicated by a minus sign. Round your answer to 2 decimal places.)

A sample of 42 observations is selected from one population with a population standard deviation of 3.1. The sample mean is 101.0. A sample of 52 observations is selected from a second population with a population standard deviation of 4.0. The sample mean is 99.8. Conduct the following test of hypothesis using the 0.04 significance level.

Explanation / Answer

Using sample data, we calculate the standard deviation (?) and compute the z-score test statistic (z).

? = sqrt[ P * ( 1 - P ) / n ] = sqrt [(0.8 * 0.2) / 100] = sqrt(0.0016) = 0.04
z = (p - P) / ? = (.73 - .80)/0.04 = -1.75

where P is the hypothesized value of population proportion in the null hypothesis, p is the sample proportion, and n is the sample size.

The value 0.05 takes on mystical characteristics. A good analyst also knows that 0.05 is not a magic number. There is little difference between P=0.04 and P=0.06. Construct a few 94 and 96% CIs from the same data set and see how little they differ (one is about 10% longer than the other.) Significance tests by themselves suggest that the research question is only about whether there is a difference, no matter the size or direction. The confidence intervals corresponding to P=0.04 and P=0.06 will probably show much the same thing, namely, they will rule out differences of practical importance in a particular direction. They will also suggest that, if there is a difference, it may or may not be of practical importance.

. Reject H0 if Z stat < -2.05 or Z stat > + 2.05

In a two-tail test, you split up alpha. So, in this case, you need .04/2 = .02 in each tail. The z-score that corresponds to 2% in the tail is z = 2.05, so A is the correct choice.

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