sample_size = random.randint(5,15)
RandomData(groups = 1, n = sample_size).z_test()3 Hypothesis Testing with z-Scores
Sample Problems
3.1 About
the one-sample z-Test problems use the RamdomData class which requires:
- the groups variable set to 1:
groups = 1 - an intger for the sample size (e.g.,
n = 30) - a call to the
z_test()method
In these sample problems, the per-group sample size is randomly set between 5 and 15. An example funciton call is included below.
3.2 Problem 1
Given the following data, is the mean of \(Group_A\) significantly different from the population mean: \(\mu = {22}\)?
Use a \({2}\) tailed-test with \(\alpha = {0.01}\)
| A |
|---|
| 32 |
| 29 |
| 25 |
| 21 |
| 35 |
| 33 |
| 31 |
| 31 |
| 34 |
The necessary summary statistics for these data
\[M_A = {30.11}\] \[{\sigma} = {4}\] \[n = {9}\]
State the Hypotheses
\[H_0: \mu = 22\] \[H_1: \mu \ne 22\]
The decision criteria:
\(z_{crit} = \pm{2.58}, \alpha_{two-tailed} = {0.01}\)
Calculate the standard error
\[\sigma_M = \frac{\sigma}{\sqrt{N}}\]
\[\sigma_M = \frac{4}{\sqrt{9}}\]
\[\sigma_M = \frac{4}{3.0}\]
\[\sigma_M = {1.33}\]
Calculate \(z_{obt}\)
\[z_{obt} = {\frac{M - \mu}{\sigma_M}}\]
\[z_{obt} = \frac{30.11 - 22}{1.33}\]
\[z_{obt} = \frac{8.11}{1.33}\]
\[z_{obt} = {6.1}\]
Calculate Cohen’s d for effect size
\[d = \frac{M - \mu}{\sigma}\]
\[d = \frac{30.11 - 22}{4}\]
\[d = \frac{8.11}{4}\]
\[d = {2.03}\]
The results:
reject the null hypothesis, results are significant,
z = 6.1, p < 0.01, d = 2.03
3.3 Problem 2
Given the following data, is the mean of \(Group_A\) significantly different from the population mean: \(\mu = {35}\)?
Use a \({2}\) tailed-test with \(\alpha = {0.05}\)
| A |
|---|
| 36 |
| 33 |
| 29 |
| 41 |
| 37 |
| 36 |
| 44 |
| 40 |
| 31 |
| 49 |
| 40 |
| 41 |
| 30 |
| 41 |
| 37 |
The necessary summary statistics for these data
\[M_A = {37.67}\] \[{\sigma} = {5}\] \[n = {15}\]
State the Hypotheses
\[H_0: \mu = 35\] \[H_1: \mu \ne 35\]
The decision criteria:
\(z_{crit} = \pm{1.96}, \alpha_{two-tailed} = {0.05}\)
Calculate the standard error
\[\sigma_M = \frac{\sigma}{\sqrt{N}}\]
\[\sigma_M = \frac{5}{\sqrt{15}}\]
\[\sigma_M = \frac{5}{3.87}\]
\[\sigma_M = {1.29}\]
Calculate \(z_{obt}\)
\[z_{obt} = {\frac{M - \mu}{\sigma_M}}\]
\[z_{obt} = \frac{37.67 - 35}{1.29}\]
\[z_{obt} = \frac{2.67}{1.29}\]
\[z_{obt} = {2.07}\]
Calculate Cohen’s d for effect size
\[d = \frac{M - \mu}{\sigma}\]
\[d = \frac{37.67 - 35}{5}\]
\[d = \frac{2.67}{5}\]
\[d = {0.53}\]
The results:
reject the null hypothesis, results are significant,
z = 2.07, p < 0.05, d = 0.53
3.4 Problem 3
Given the following data, is the mean of \(Group_A\) significantly different from the population mean: \(\mu = {37}\)?
Use a \({2}\) tailed-test with \(\alpha = {0.01}\)
| A |
|---|
| 26 |
| 66 |
| 27 |
| 56 |
| 46 |
| 61 |
| 49 |
| 50 |
| 53 |
| 60 |
| 45 |
| 39 |
| 47 |
The necessary summary statistics for these data
\[M_A = {48.08}\] \[{\sigma} = {10}\] \[n = {13}\]
State the Hypotheses
\[H_0: \mu = 37\] \[H_1: \mu \ne 37\]
The decision criteria:
\(z_{crit} = \pm{2.58}, \alpha_{two-tailed} = {0.01}\)
Calculate the standard error
\[\sigma_M = \frac{\sigma}{\sqrt{N}}\]
\[\sigma_M = \frac{10}{\sqrt{13}}\]
\[\sigma_M = \frac{10}{3.61}\]
\[\sigma_M = {2.77}\]
Calculate \(z_{obt}\)
\[z_{obt} = {\frac{M - \mu}{\sigma_M}}\]
\[z_{obt} = \frac{48.08 - 37}{2.77}\]
\[z_{obt} = \frac{11.08}{2.77}\]
\[z_{obt} = {4.0}\]
Calculate Cohen’s d for effect size
\[d = \frac{M - \mu}{\sigma}\]
\[d = \frac{48.08 - 37}{10}\]
\[d = \frac{11.08}{10}\]
\[d = {1.11}\]
The results:
reject the null hypothesis, results are significant,
z = 4.0, p < 0.01, d = 1.11
3.5 Problem 4
Given the following data, is the mean of \(Group_A\) significantly different from the population mean: \(\mu = {27}\)?
Use a \({2}\) tailed-test with \(\alpha = {0.05}\)
| A |
|---|
| 45 |
| 40 |
| 45 |
| 49 |
| 39 |
| 40 |
| 41 |
| 41 |
The necessary summary statistics for these data
\[M_A = {42.5}\] \[{\sigma} = {5}\] \[n = {8}\]
State the Hypotheses
\[H_0: \mu = 27\] \[H_1: \mu \ne 27\]
The decision criteria:
\(z_{crit} = \pm{1.96}, \alpha_{two-tailed} = {0.05}\)
Calculate the standard error
\[\sigma_M = \frac{\sigma}{\sqrt{N}}\]
\[\sigma_M = \frac{5}{\sqrt{8}}\]
\[\sigma_M = \frac{5}{2.83}\]
\[\sigma_M = {1.77}\]
Calculate \(z_{obt}\)
\[z_{obt} = {\frac{M - \mu}{\sigma_M}}\]
\[z_{obt} = \frac{42.5 - 27}{1.77}\]
\[z_{obt} = \frac{15.5}{1.77}\]
\[z_{obt} = {8.76}\]
Calculate Cohen’s d for effect size
\[d = \frac{M - \mu}{\sigma}\]
\[d = \frac{42.5 - 27}{5}\]
\[d = \frac{15.5}{5}\]
\[d = {3.1}\]
The results:
reject the null hypothesis, results are significant,
z = 8.76, p < 0.05, d = 3.1
3.6 Problem 5
Given the following data, is the mean of \(Group_A\) significantly different from the population mean: \(\mu = {22}\)?
Use a \({2}\) tailed-test with \(\alpha = {0.01}\)
| A |
|---|
| 30 |
| 29 |
| 30 |
| 30 |
| 31 |
The necessary summary statistics for these data
\[M_A = {30.0}\] \[{\sigma} = {2}\] \[n = {5}\]
State the Hypotheses
\[H_0: \mu = 22\] \[H_1: \mu \ne 22\]
The decision criteria:
\(z_{crit} = \pm{2.58}, \alpha_{two-tailed} = {0.01}\)
Calculate the standard error
\[\sigma_M = \frac{\sigma}{\sqrt{N}}\]
\[\sigma_M = \frac{2}{\sqrt{5}}\]
\[\sigma_M = \frac{2}{2.24}\]
\[\sigma_M = {0.89}\]
Calculate \(z_{obt}\)
\[z_{obt} = {\frac{M - \mu}{\sigma_M}}\]
\[z_{obt} = \frac{30.0 - 22}{0.89}\]
\[z_{obt} = \frac{8.0}{0.89}\]
\[z_{obt} = {8.99}\]
Calculate Cohen’s d for effect size
\[d = \frac{M - \mu}{\sigma}\]
\[d = \frac{30.0 - 22}{2}\]
\[d = \frac{8.0}{2}\]
\[d = {4.0}\]
The results:
reject the null hypothesis, results are significant,
z = 8.99, p < 0.01, d = 4.0