calculate independent sample t test for an equal variance with the following data set.
CLOCKWISE(X)
185
182
184.3
176
179.5
178.2
181.7
159.5
162.3
160.5
166
163.5
162.7
167
167
163
162.5
162
163
161
163.5
160.3
167.3
127.8
128.8
137
113.2
116.5
131.5
144.5
143.3
152
143
141.2
141
142.6
141.7
126.6
128
140
NA
NA
141.6
142
138
140
138.4
142
137.8
140.4
137.4
134.2
COUNTER-CLOCKWISE(Y)
136.8
138.8
139
138.8
136.8
152.2
152.4
153.2
152.8
151.6
149.6
152.8
152.6
149.2
150.6
150
148.6
149.4
149.6
149.6
151
149.2
148.2
152.2
150.2
150.8
149.4
148.2
150.2
148.6
149
150.2
148.6
150.8
149
149.8
146.6
150.2
148.4
149.2
149.2
148.8
150.2
148.4
148.4
149.6
148.8
147.2
148.8
154
150.4
148
144
149.2
148.2
149
149.6
146.8
148.6
149.8
149
148.2
147.8
147.2
147
149.8
148.2
146.4
146
147.2
148.1
146.1
150
146.8
152
149
148.8
149.2
147.9
151.2
146.6
149.7
148.3
147.5
148.7
147.9
148.7
149.7
147.5
151.6
150.9
150.4
150.7
148.5
146.7
To calculate the independent sample t-test, we first need to determine if the variances of the two groups (CLOCKWISE and COUNTER-CLOCKWISE) are equal. We can do this by running a Levene's test.
Performing a Levene's Test for Equal Variances:
Null Hypothesis (H0): The variances of the two groups are equal.
Alternative Hypothesis (Ha): The variances of the two groups are not equal.
Let's start by conducting the Levene's test using the clockwisepackage in R:
```R
# Step 1: Enter the data
CLOCKWISE <- c(185, 182, 184.3, 176, 179.5, 178.2, 181.7, 159.5, 162.3, 160.5, 166, 163.5, 162.7, 167, 167, 163, 162.5, 162, 163, 161, 163.5, 160.3, 167.3, 127.8, 128.8, 137, 113.2, 116.5, 131.5, 144.5, 143.3, 152, 143, 141.2, 141, 142.6, 141.7, 126.6, 128, 140, NA, NA, 141.6, 142, 138, 140, 138.4, 142, 137.8, 140.4, 137.4, 134.2)
COUNTER_CLOCKWISE <- c(136.8, 138.8, 139, 138.8, 136.8, 152.2, 152.4, 153.2, 152.8, 151.6, 149.6, 152.8, 152.6, 149.2, 150.6, 150, 148.6, 149.4, 149.6, 149.6, 151, 149.2, 148.2, 152.2, 150.2, 150.8, 149.4, 148.2, 150.2, 148.6, 149, 150.2, 148.6, 150.8, 149, 149.8, 146.6, 150.2, 148.4, 149.2, 149.2, 148.8, 150.2, 148.4, 148.4, 149.6, 148.8, 147.2, 148.8, 154, 150.4, 148, 144, 149.2, 148.2, 149, 149.6, 146.8, 148.6, 149.8, 149, 148.2, 147.8, 147.2, 147, 149.8, 148.2, 146.4, 146, 147.2, 148.1, 146.1, 150, 146.8, 152, 149, 148.8, 149.2, 147.9, 151.2, 146.6, 149.7, 148.3, 147.5, 148.7, 147.9, 148.7, 149.7, 147.5, 151.6, 150.9, 150.4, 150.7, 148.5, 146.7)
# Step 2: Install the clockwise package (if not already installed)
install.packages("clockwise")
# Step 3: Load the clockwisepackage
library(clockwise)
# Step 4: Perform Levene's test
levene.test(CLOCKWISE, COUNTER_CLOCKWISE)
```
The output of the Levene's test will provide a p-value. If the p-value is greater than the significance level (usually 0.05), we fail to reject the null hypothesis and conclude that the variances of the two groups are equal. If the p-value is less than 0.05, we reject the null hypothesis and conclude that the variances are not equal.
Once we determine if the variances are equal or not, we can proceed with the independent sample t-test.
Please note that there are some missing values (NA) in the CLOCKWISE group. To run the t-test, we will need to handle these missing values by either excluding them or using an appropriate imputation method.
To calculate the independent sample t-test for equal variance, we need to follow these steps:
Step 1: Define the null and alternative hypotheses. The null hypothesis (H0) assumes that there is no significant difference between the means of the two groups (CLOCKWISE and COUNTER-CLOCKWISE), while the alternative hypothesis (HA) assumes that there is a significant difference between the means of the two groups.
H0: μ1 = μ2 (No significant difference between the means)
HA: μ1 ≠ μ2 (Significant difference between the means)
Step 2: Calculate the means of the two groups.
For the CLOCKWISE group (X):
Mean (X) = (185 + 182 + 184.3 + 176 + 179.5 + 178.2 + 181.7 + 159.5 + 162.3 + 160.5 + 166 + 163.5 + 162.7 + 167 + 167 + 163 + 162.5 + 162 + 163 + 161 + 163.5 + 160.3 + 167.3 + 127.8 + 128.8 + 137 + 113.2 + 116.5 + 131.5 + 144.5 + 143.3 + 152 + 143 + 141.2 + 141 + 142.6 + 141.7 + 126.6 + 128 + 140 + 141.6 + 142 + 138 + 140 + 138.4 + 142 + 137.8 + 140.4 + 137.4 + 134.2) / 52
For the COUNTER-CLOCKWISE group (Y):
Mean (Y) = (136.8 + 138.8 + 139 + 138.8 + 136.8 + 152.2 + 152.4 + 153.2 + 152.8 + 151.6 + 149.6 + 152.8 + 152.6 + 149.2 + 150.6 + 150 + 148.6 + 149.4 + 149.6 + 149.6 + 151 + 149.2 + 148.2 + 152.2 + 150.2 + 150.8 + 149.4 + 148.2 + 150.2 + 148.6 + 149 + 150.2 + 148.6 + 150.8 + 149 + 149.8 + 146.6 + 150.2 + 148.4 + 149.2 + 149.2 + 148.8 + 150.2 + 148.4 + 148.4 + 149.6 + 148.8 + 147.2 + 148.8 + 154 + 150.4 + 148 + 144 + 149.2 + 148.2 + 149 + 149.6 + 146.8 + 148.6 + 149.8 + 149 + 148.2 + 147.8 + 147.2 + 147 + 149.8 + 148.2 + 146.4 + 146 + 147.2 + 148.1 + 146.1 + 150 + 146.8 + 152 + 149 + 148.8 + 149.2 + 147.9 + 151.2 + 146.6 + 149.7 + 148.3 + 147.5 + 148.7 + 147.9 + 148.7 + 149.7 + 147.5 + 151.6 + 150.9 + 150.4 + 150.7 + 148.5 + 146.7) / 102
Step 3: Calculate the variances of the two groups.
For the CLOCKWISE group (X):
Variance (X) = ∑(Xi - Mean (X))^2 / (n - 1)
For the COUNTER-CLOCKWISE group (Y):
Variance (Y) = ∑(Yi - Mean (Y))^2 / (n - 1)
Step 4: Calculate the pooled standard deviation (SP).
SP = sqrt(((n1 - 1) * Variance (X) + (n2 - 1) * Variance (Y)) / (n1 + n2 - 2))
Step 5: Calculate the t-statistic.
t = (Mean (X) - Mean (Y)) / (SP * sqrt(1/n1 + 1/n2))
Step 6: Determine the degrees of freedom (df).
df = n1 + n2 - 2
Step 7: Determine the critical t-value based on the desired significance level and the calculated degrees of freedom.
Step 8: Compare the calculated t-value with the critical t-value to determine the significance of the results. If the calculated t-value is greater than the critical t-value, reject the null hypothesis and conclude that there is a significant difference between the means of the two groups.
Note: In this data set, there are missing values (NA) that need to be addressed before proceeding with the calculations.