Standard Error of the Mean, Confidence Intervals, & The Z Test
Report the values you are using for this assignment.
HR Mean (µHR)
(one decimal)
HR Std Dev (sHR)
(two decimals)
RR Mean (µRR)
(one decimal)
RR Std Dev (sRR)
(two decimals)
90.7
6.83
12.6
1.96
Make sure these initial values have been marked correct.
Follow the rounding instructions carefully.
Task 1: Standard Error of the Mean (SEM) for Different Sample Sizes
Imagine that your HR and RR data sets were very large (N=500). Calculate SEM based on the population standard deviations calculated in assignment 2, and the sample sizes given in the table. Carefully round the SEM value to 2 decimals for each calculation.
Formula: Standard Error of the Mean (SEM or sM or ) =
SEM
Heart Rate
Respiratory Rate
SEM if n = 9
2.28
0.65
SEM if n = 25
1.37
0.39
SEM if n= 100
0.68
0.2
What happens to the value of the SEM as the sample size increases?
By observing the values, we can conclude that as the sample size (n) increases the standard error of the mean decreases.
Question:
Task 5: Z Test: Calculate the Z statistic (Zobt) and compare to Zcrit
*Reminder! Your data is now viewed as a sample. Use the σ given in Task 4 to calculate SEM.
1) Use 3.16 as the square root of 10 to calculate the Standard Error of the Mean.
(You may also see SEM expressed as sM or )
(SEM or sM or ) = Round to 2 decimals.
2) Calculate the difference between the means (sample minus population)
!! If the difference is zero, use 1.1 as the difference.
3) Calculate ZOBT using:
HR
SEM
ZOBT
Value
90.7 – 81.36
zHR = 2.08
RR
SEM
ZOBT
Value
12.6 – 15.69
zRR = -2.54
Alpha level (α) for all test is 0.05. The Zcrit values are:
Two Tailed
-1.96 and +1.96 (or ±1.96)
One Tailed
If difference between means is negative: -1.65
If difference between means is positive: +1.65
4) Compare ZOBT to ZCRIT. Is ZOBT located in the Zone of Rejection? (aka the Critical Region)
Sketching a number line with critical values and zones identified helps with this.
Zone of Zone of
Rejection -----------------------------Do Not Reject--------------------------------- Rejection
-1.96 0 1.96
Use the same principle for the one tailed comparison. In that case there will only be one Zone of Rejection, based on whether Zcrit was positive or negative.
Task 6: Z Test: Decisions, Effect Size, and Potential Error Type
1) Answer the first two questions (Yes or No) based on what you found in Task 2.
2) Calculate effect size using Cohen’s d: d = (Reported as a positive value.)
(Effect size is not usually reported if the null is not rejected, but do it here just to build skills.)
3) There is always a theoretical statistical error potentially present after a hypothesis is tested. Name the potential error that must be acknowledged because of your decision.
HR
Two Tail
One Tail
Is your decision to Reject the Null?
Can you say the sample mean is statistically different from the population mean?
What is the effect size? (Cohen’s d)
1.59
What type of statistical error might be present?
RR
Two Tail
One Tail
Is your decision to Reject the Null?
Can you say the sample mean is statistically different from the population mean?
What is the effect size? (Cohen’s d)
What type of statistical error might be present?
→ Consult the notes for detailed information on one and two tailed tests, hypothesis construction, alpha level, critical values, Zone of Rejection, types of error, and effect size by Cohen’s d.
→ The GWFW book uses the term Z ‘score’ even when comparing means; perhaps to highlight the similarity between the formula for a Z score (with sigma ‘σ’ in the denominator) and the formula for the Z statistic (with SEM in the denominator.) At the Z test level, a statistical test for comparing means is being performed by calculating a Z statistic. When the context is clear, ‘Z value’, or just ‘Z’, could be used as a general term of reference, but you should be aware that a Z score is not the same as a Z statistic.
Standard Error of the Mean, Confidence Intervals, & The Z Test
Report the values you are using for this assignment.
HR Mean (µHR)
(one decimal)
HR Std Dev (sHR)
(two decimals)
RR Mean (µRR)
(one decimal)
RR Std Dev (sRR)
(two decimals)
90.7
6.83
12.6
1.96
Make sure these initial values have been marked correct.
Follow the rounding instructions carefully.
Task 1: Standard Error of the Mean (SEM) for Different Sample Sizes
Imagine that your HR and RR data sets were very large (N=500). Calculate SEM based on the population standard deviations calculated in assignment 2, and the sample sizes given in the table. Carefully round the SEM value to 2 decimals for each calculation.
Formula: Standard Error of the Mean (SEM or sM or ) =
SEM
Heart Rate
Respiratory Rate
SEM if n = 9
2.28
0.65
SEM if n = 25
1.37
0.39
SEM if n= 100
0.68
0.2
What happens to the value of the SEM as the sample size increases?
By observing the values, we can conclude that as the sample size (n) increases the standard error of the mean decreases.
Question:
Task 5: Z Test: Calculate the Z statistic (Zobt) and compare to Zcrit
*Reminder! Your data is now viewed as a sample. Use the σ given in Task 4 to calculate SEM.
1) Use 3.16 as the square root of 10 to calculate the Standard Error of the Mean.
(You may also see SEM expressed as sM or )
(SEM or sM or ) = Round to 2 decimals.
2) Calculate the difference between the means (sample minus population)
!! If the difference is zero, use 1.1 as the difference.
3) Calculate ZOBT using:
HR
SEM
ZOBT
Value
90.7 – 81.36
zHR = 2.08
RR
SEM
ZOBT
Value
12.6 – 15.69
zRR = -2.54
Alpha level (α) for all test is 0.05. The Zcrit values are:
Two Tailed
-1.96 and +1.96 (or ±1.96)
One Tailed
If difference between means is negative: -1.65
If difference between means is positive: +1.65
4) Compare ZOBT to ZCRIT. Is ZOBT located in the Zone of Rejection? (aka the Critical Region)
Sketching a number line with critical values and zones identified helps with this.
Zone of Zone of
Rejection -----------------------------Do Not Reject--------------------------------- Rejection
-1.96 0 1.96
Use the same principle for the one tailed comparison. In that case there will only be one Zone of Rejection, based on whether Zcrit was positive or negative.
Task 6: Z Test: Decisions, Effect Size, and Potential Error Type
1) Answer the first two questions (Yes or No) based on what you found in Task 2.
2) Calculate effect size using Cohen’s d: d = (Reported as a positive value.)
(Effect size is not usually reported if the null is not rejected, but do it here just to build skills.)
3) There is always a theoretical statistical error potentially present after a hypothesis is tested. Name the potential error that must be acknowledged because of your decision.
HR
Two Tail
One Tail
Is your decision to Reject the Null?
Can you say the sample mean is statistically different from the population mean?
What is the effect size? (Cohen’s d)
1.59
What type of statistical error might be present?
RR
Two Tail
One Tail
Is your decision to Reject the Null?
Can you say the sample mean is statistically different from the population mean?
What is the effect size? (Cohen’s d)
What type of statistical error might be present?
→ Consult the notes for detailed information on one and two tailed tests, hypothesis construction, alpha level, critical values, Zone of Rejection, types of error, and effect size by Cohen’s d.
→ The GWFW book uses the term Z ‘score’ even when comparing means; perhaps to highlight the similarity between the formula for a Z score (with sigma ‘σ’ in the denominator) and the formula for the Z statistic (with SEM in the denominator.) At the Z test level, a statistical test for comparing means is being performed by calculating a Z statistic. When the context is clear, ‘Z value’, or just ‘Z’, could be used as a general term of reference, but you should be aware that a Z score is not the same as a Z statistic.
