Seconds Speed
144 37.5
149.2 36.193029
148.2 36.437247
149 36.241611
149.6 36.096257
146.8 36.784741
148.6 36.339166
149.8 36.048064
149 36.241611
148.2 36.437247
147.8 36.535859
147.2 36.684783
147 36.734694
149.8 36.048064
148.2 36.437247
Calculate independent sample t test for equal variance using the data above?
To calculate the independent sample t-test for equal variance using the data above, we need to compare the means of the two groups.
First, we calculate the mean of each group:
Group 1 mean = (37.5 + 36.241611 + 36.096257 + 36.241611 + 36.241611) / 5 = 36.6240208
Group 2 mean = (36.193029 + 36.437247 + 36.784741 + 36.339166 + 36.048064 + 36.437247 + 36.535859 + 36.684783 + 36.734694 + 36.048064 + 36.437247) / 11 = 36.44720636
Next, we need to calculate the variance of each group:
Group 1 variance = ((37.5 - 36.6240208)^2 + (36.241611 - 36.6240208)^2 + (36.096257 - 36.6240208)^2 + (36.241611 - 36.6240208)^2 + (36.241611 - 36.6240208)^2) / 4 = 1.000349864
Group 2 variance = ((36.193029 - 36.44720636)^2 + (36.437247 - 36.44720636)^2 + (36.784741 - 36.44720636)^2 + (36.339166 - 36.44720636)^2 + (36.048064 - 36.44720636)^2 + (36.437247 - 36.44720636)^2 + (36.535859 - 36.44720636)^2 + (36.684783 - 36.44720636)^2 + (36.734694 - 36.44720636)^2 + (36.048064 - 36.44720636)^2 + (36.437247 - 36.44720636)^2) / 10 = 0.056577453
(Note: The variance formula used here is the sample variance formula, dividing by n-1 instead of n.)
With these values, we can now calculate the t-statistic using the formula:
t = (mean1 - mean2) / sqrt((variance1/5) + (variance2/11))
= (36.6240208 - 36.44720636) / sqrt((1.000349864/5) + (0.056577453/11))
= 0.17681444 / sqrt(0.200069973 + 0.00514341)
= 0.17681444 / sqrt(0.205213383)
Finally, we can compare the calculated t-statistic to the critical value to determine if there is a significant difference between the two groups. The critical value depends on the desired level of significance and the degrees of freedom (n1-1 + n2-1 - 2).
To calculate an independent sample t-test for equal variance using the data provided, follow these steps:
Step 1: Define the null and alternative hypotheses.
- Null Hypothesis (H0): There is no significant difference between the means of the two groups.
- Alternative Hypothesis (Ha): There is a significant difference between the means of the two groups.
Step 2: Calculate the mean and standard deviation for each group.
- Group 1 (Speed): Calculate the mean and standard deviation for the first column of data.
- Mean 1 = (144 + 149.2 + 148.2 + 149.6 + 146.8 + 148.6 + 149 + 149 + 148.2 + 147.8 + 147.2 + 147 + 149.8 + 148.2) / 14
- Mean 1 = 147.992857143
- Standard Deviation 1 = √(Σ(x - mean)² / (n - 1))
- Group 2 (Seconds): Calculate the mean and standard deviation for the second column of data.
- Mean 2 = (37.5 + 36.193029 + 36.437247 + 36.241611 + 36.096257 + 36.784741 + 36.339166 + 36.048064 + 36.241611 + 36.437247 + 36.535859 + 36.684783 + 36.734694 + 36.048064) / 14
- Mean 2 = 36.53679679
- Standard Deviation 2 = √(Σ(x - mean)² / (n - 1))
Step 3: Calculate the test statistic.
- Use the formula: t = (mean1 - mean2) / √((sd1² / n1) + (sd2² / n2))
- Plug in the values from Step 2: t = (147.992857143 - 36.53679679) / √((Standard Deviation 1² / 14) + (Standard Deviation 2² / 14))
Step 4: Calculate the degrees of freedom (df).
- Degrees of Freedom = n1 + n2 - 2
- In this case, both groups have 14 observations, so df = 14 + 14 - 2 = 26.
Step 5: Determine the critical value and p-value.
- Look up the critical value for the appropriate significance level and degrees of freedom using a t-distribution table.
- Calculate the p-value associated with the test statistic using software or a t-distribution table.
Step 6: Make a decision.
- If the test statistic is greater than the critical value or the p-value is less than the chosen significance level (e.g., α = 0.05), reject the null hypothesis. There is sufficient evidence to support the alternative hypothesis.
- If the test statistic is less than the critical value or the p-value is greater than the chosen significance level, fail to reject the null hypothesis. There is not enough evidence to support the alternative hypothesis.
Please note that the calculation steps provided are simplified and may require rounding or additional adjustments based on the specific requirements of the statistical software or methodology being used.