Sum of Squares (SS)
What is the Sum of Squares (SS)
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.
Task 2: Confidence Intervals
Use the SEM calculated in Task 1 to build a 95% Confidence interval for a sample of n=25. First table is for the prep calculations. Round the Z*SEM value to 2 decimals. Use the rounded value for the final calculations.
Second table displays the final values somewhat visually: you could picture a normal curve and number line above the table. This may help you decide whether the sample means mentioned would be expected to fall within the 95% CI based on your population mean.
95% Confidence Intervals
SEM for n=25
Z for 95% CI
Z * SEM
12.6
1.96
0.392
Lower Limit
µ - [Z*SEM]
Mean
Upper Limit
µ + [Z*SEM]
11.83
12.6
13.37
Would a sample mean of 81.7 fall within the 95% CI for HR?
A sample mean of 81.7 would not fall within this interval, which suggests that the true population means may be different from the observed sample Mean. For RR, we don’t have the population mean or standard deviation, so we can only calculate a one-sided 95% confidence interval.
SEM for n=25
Z for 95% CI
Z * SEM
12.6
1.96
0.392
Lower Limit
µ - [Z*SEM]
Mean
Upper Limit
µ + [Z*SEM]
12.6
12.6
0.77
Would a sample mean of 14.2 fall within the 95% CI for RR?
No, a mean of 14.2 will not fall within 95% CI for RR.
99% Confidence Interval for HR
SEM for n=25
Z for 99% CI
Z * SEM
1.37
2.576
Lower Limit
µ - [Z*SEM]
Mean
Upper Limit
µ + [Z*SEM]
86.74
90.7
94.66
Would a sample mean of 81.7 fall within the 95% CI for HR?
The sample mean of 81.7 would not fall within the 95% confidence interval for HR since the lower limit of the 99% confidence interval is 83.00 which is higher than the given sample mean of 81.7
SEM for n=25
Z for 99% CI
Z * SEM
11.59
12.6
13.61
Lower Limit
µ - [Z*SEM]
Mean
Upper Limit
µ + [Z*SEM]
60.29
83.94
105.71
Would a sample mean of 14.2 fall within the 95% CI for RR?
No, as it does not contain the confidence interval.
What happens to the width of the interval if the confidence level goes from 95% to 99%?
No, as it does not contain the conference Interval.
What would happen to the width of the interval if the sample size was n=9?
As the margin of error decreases when sample size increases. Hence, the confidence interval becomes wider.
What would happen to the width of the interval if the sample size was n=100?
If n = 100 then, the 95% confidence interval is given by; = xbar - + tc*(sigma/sqrt(n))
= 12.6 -+ 1.96*(1.96/sqrt(100)) C.I (12.09,13.10). As the margin of error decreases when sample size increases. Hence, the confidence interval becomes narrower.
Task 3: Sample Standard Deviation (i.e., unbiased estimate of s)
Samples are usually less variable than the population they came from, so we say they are ‘biased’ toward lower variability. ‘Bessel’s correction’ adjusts for this bias: when calculating the variance, the sum of squares is divided by the number of scores minus 1 (n-1). This makes the variance a bit larger. The smaller the sample, the larger the effect of the correction – which is ideal, because the smallest samples would have the most bias.
The unbiased estimate of the population standard deviation (aka the ‘Sample Standard Deviation’) is given the symbol ‘s’. The population symbol is sigma (s), which is a Greek ‘s’.
Treat your 10 HR measures as a sample, and calculate the sample standard deviation (s.) The Sum of Squares (SS) calculated in Assignment 2 is still valid, of course, but now ‘variance’ or refers to the unbiased variance calculated using (n-1.)
HR
RR
Sum of Squares (SS)
Using
the sample Variance is:
51.93
4.37
Using s
the unrounded sample standard deviation is
6.83
1.96
Rounded to 2 decimals,
the sample Standard Deviation (s) is
7.21
2.09
Error check: Is your sample standard deviation greater than your sigma? (it should be)
yes
yes
→ Although you have learned about the population standard deviation first, it’s really the sample standard deviation that is more commonly used in research. Rather than say ‘unbiased variance’ or ‘unbiased estimate of the population standard deviation’, people just say ‘variance’, ‘sample standard deviation’ or ‘standard deviation’ and it is understood that they are talking about sample measures. They may say ‘sigma’ to indicate the population standard deviation.
→ In this course we use the Z table, which requires a population standard deviation (sigma). If you only have the standard deviation of a sample, you use a different table to find the locations associated with particular locations and proportions. It’s called a ‘t table’ and it’s likely to be covered in your next statistics course.
Task 4: Z Test: Null Hypothesis and Alternate Hypothesis
For the Z test, view the mean of your 10 measures ( ME) as the mean of a random sample of n=10 taken from a large population. Just as a teacher might compare the mean of their class to the provincial mean, you will compare your mean to a population mean, using a Z test.
Begin by reporting your means (with 1 decimal.) Population values are provided.
Heart Rate
Respiratory Rate
me
90.7
12.6
µpop
81.36
15.69
spop
14.21
3.84
The hypothesis statements are partially set up in the table. Your task is to supply the symbol that completes each hypothesis statement (i.e., whatever should replace the ‘?’)
Use mathematical symbols if your technology supports that.
Use the words or phrases below if symbols are not available to you.
= (is equal to) ≤ (is less than or equal to) ≥ (is greater than or equal to)
≠ (is not equal to) > (is greater than) < (is less than)
→ For the one tailed test, if your mean is numerically less than the population mean, choose a null and alternate that are appropriate for testing if your mean is statistically less. (Same idea if greater.)
Hypothesis
Statement
HR Two Tailed
HR One Tailed
Null
H0: µme ? 81.36
81.36
81.36
Alternate
HA: µme ? 81.36
81.36
81.36
Hypothesis
Statement
RR Two Tailed
RR One Tailed
Null
H0: µme ? 15.69
15.69
15.69
Alternate
HA: µme ? 15.69
15.69
15.69
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
2.16
4.32
zHR = 2.08
RR
SEM
ZOBT
Value
12.6 – 15.69
0.2
-15.45
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?
Yes, the decision is to reject the null hypothesis.
Yes, the decision is to reject the null hypothesis.
Can you say the sample mean is statistically different from the population mean?
Yes
Yes
What is the effect size? (Cohen’s d)
1.59
1.59
What type of statistical error might be present?
Type I error (rejecting the null hypothesis when it is actually true) might be present.
Type I error (rejecting the null hypothesis when it is actually true) might be present.
RR
Two Tail
One Tail
Is your decision to Reject the Null?
Yes, the decision is to reject the null hypothesis.
Yes, the decision is to reject the null hypothesis.
Can you say the sample mean is statistically different from the population mean?
Yes
Yes
What is the effect size? (Cohen’s d)
The effect size is not reported as the instructions state it is not usually reported if the null is not rejected.
The effect size is not reported as the instructions state it is not usually reported if the null is not rejected.
What type of statistical error might be present?
Type I (rejecting the null hypothesis when it is actually true) might be present.
Type I (rejecting the null hypothesis when it is actually true) 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.
