The Z Test

For this assignment, view the mean of your 10 measures (_me) as the mean of a random sample of n=10 taken from a large population. This is analogous to a teacher comparing the mean of one class to the mean of all students in the district. Begin by reporting your means (they have only 1 decimal.)

	Heart Rate	Respiratory Rate
me	90	23
pop	80.02	15.08
spop	19.47	4.65

The Z test is conceptually related to Confidence Intervals. The difference is that instead of calculating upper and lower limits and seeing whether the mean you are interested in falls within those score values, you just use the relevant Z values as the limits (call them Z_crit) and use a formula to convert the mean you are interested in to a Z statistic (Z_obt). Then you see whether Z_obt falls within the large area near the mean of the Z distribution (0), or the extreme area in the tail(s).

Task 1: Hypothesis statements and preparation

1) Based on the wording of the question, decide if the test is one tailed or two tailed, and write formal hypothesis statements.

2) The number of tails tells you how many rejection zones you have, and the alpha level tells you what proportion of the normal curve is in the rejection zone(s). Find the Z that defines that proportion. That is the 'Critical Value' (or values). Be mindful of + and/or - signs!

Task 2: Calculate the obtained value of Z for the Z test

1) Calculate the difference between the means (sample minus population)

2) Using 3.16 as the square root of 10, calculate the Standard Error of the Mean using:

(SEM ors_M or)= Round to 2 decimals.

3) Calculate Z_obt using:

Task 3: Decisions, Statistical Error, Effect Size, and Summary

1) Each of the four statements can be answered in only two ways. The way you answer the first statement determines the next three answers. Consult the notes carefully and remember that 'extreme' means 'far away from the center'.

2) Calculate effect size using Cohen's d:(Reported as a positive value.)

3) Write a formal summary. (Describe the analysis, stating the means, their sources, type of comparison (higher, lower, different), and n of sample. Mention the type of statistical test performed, including the alpha level. Compare the Z_obt and Z_crit values, and whether this meant rejection of the null hypothesis. State the effect size, and acknowledge the potential for statistical error. Effect size is not usually reported if the null is not rejected, but do it here just to build skills.)

Task 1: Z test for Heart Rate

Use the Z test to determine whether the mean of your sample of 10 HR measurements is statistically different from mean of a larger population.

Use 3.16 as the square root of 10. Use an alpha level of .05.

Write your formal null and alternate hypothesis statements using mathematical symbols, then complete the table.

Null:

Alternate:

Is this a one tailed or two tailed question?
(Zcrit) The Critical Value(s) of Z
	9.98
SEM	1.95
Zobt=/SEM Zobt	5.12
Is Zobtmore extreme or less extreme than Zcrit?
Is your decision to Reject or Not Reject the Null
Would you report the sample mean is or is not statistically different from the population mean?
What type of statistical error might be present?
d=/s What is the value of Cohen's d?

Summarize your findings in a formal statement.

Task 2: Z test for Respiratory Rate

If the mean of your 10 measures is mathematically lower than the population mean, use the Z test to decide whether it is statistically lower.

If the mean of your 10 measures is mathematically higher than the population mean, use the Z test to decide whether it is statistically higher.

Use 3.16 as the square root of 10. Use an alpha level of .05.

Write your formal null and alternate hypothesis statements, using mathematical symbols, then complete the table.

Null:

Alternate:

Is this a one tailed or two tailed question?
(Zcrit) The Critical Value of Z is
is	7.92
SEM is
Zobt= (/SEM Zobt is
Is Zobtmore extreme or is it less extreme than Zcrit?
Is your decision to Reject or Not Reject the Null
Would you report the mean of your 10 measures is or is not statistically higher (lower) than the population mean?
What type of statistical error might be present?
d= (/s What is the value of Cohen's d?

Summarize your findings in a formal statement. (See #3 of Task 3 in instructions)