14) You wish to compare the average GPA of students from one department to another. It is assumed that the distributions of GPA in both departments are normally distributed and that the overall standard deviation of GPA is the same in both departments. If you sample 17 students from the first department and 25 students from the second department, what is the correct (Z or t) multiplier for a 95% Confidence Interval on the mean difference between the 2 departments?
To determine the correct multiplier for a 95% Confidence Interval (CI) on the mean difference between the two departments, we need to consider the sample sizes of both departments.
For sample sizes larger than 30, it is common to use the z-distribution, which is appropriate when the population standard deviation is known. However, in this case, the population standard deviation is not known. Therefore, we need to use the t-distribution, which is appropriate when the population standard deviation is unknown and estimated from the sample.
To calculate the multiplier for a 95% Confidence Interval using the t-distribution, we need to find the degrees of freedom (df). The degrees of freedom for comparing the means of two independent groups is given by the formula:
df = (n1 + n2) - 2
Where:
n1 = sample size of the first department (17 in this case)
n2 = sample size of the second department (25 in this case)
In this example:
df = (17 + 25) - 2
df = 40 - 2
df = 38
To find the correct multiplier for a 95% Confidence Interval using the t-distribution with 38 degrees of freedom, you can refer to a t-table or use statistical software. The multiplier for a 95% Confidence Interval with 38 degrees of freedom is approximately 2.022.
Therefore, the correct multiplier for a 95% Confidence Interval on the mean difference between the two departments is approximately 2.022, using the t-distribution.
To calculate the correct multiplier for a 95% Confidence Interval on the mean difference between the two departments, we need to determine whether to use a Z or t distribution.
Since the sample sizes for both departments are relatively large (17 and 25), we can use the Z-distribution. Generally, when sample sizes are greater than or equal to 30, the t-distribution approximation is no longer necessary.
Therefore, to calculate the correct multiplier for a 95% Confidence Interval, we will use the Z-distribution.
The Z-multiplier for a 95% Confidence Interval is approximately 1.96.
Therefore, the correct Z-multiplier for a 95% Confidence Interval on the mean difference between the two departments is 1.96.