You want to estimate the difference in grade point averages between two groups of university
students to be accurate within 0.2 grade point, with probability approximately equal to 0.95. If the
standard deviation of the grade point measurements is approximately equal to 0.6, how many students
must be included in each group?
[(1.96^2)(0.6^2)]/(0.2^2)
= 34.5744
~35 students must be included in each group
To estimate the required sample size for each group, we can use the formula for sample size calculation in estimating the difference between two means. Here's the step-by-step explanation:
Step 1: Identify the given information:
- Desired margin of error (E): 0.2 grade point (the maximum difference that can be tolerated)
- Confidence level: approximately 0.95 (we want a probability of 95% that the estimated value falls within the specified range)
- Standard deviation (σ): 0.6 grade point (the variability in the data)
Step 2: Determine the critical value (Z-score) corresponding to the desired confidence level. In this case, since the confidence level is approximately 0.95, we need to find the Z-score that corresponds to a 95% confidence level. This value can be obtained from a standard normal distribution table or using a statistical software. Let's assume the critical value is Z.
Step 3: Use the formula to calculate the required sample size (n):
n = (Z^2 * σ^2) / E^2
Substituting the given values:
n = (Z^2 * σ^2) / E^2
n = (Z^2 * 0.6^2) / 0.2^2
Step 4: Calculate the sample size.
Now, to find the sample size for each group, we need to calculate the value of Z and substitute it into the formula:
n = (Z^2 * 0.6^2) / 0.2^2
The value of Z depends on the desired level of confidence. For a 95% confidence level, the Z-score is approximately 1.96 (as obtained from a Z-table or statistical software).
n = (1.96^2 * 0.6^2) / 0.2^2
n = (3.8416 * 0.36) / 0.04
n = 13.8912 / 0.04
n ≈ 347.28
Since the sample size must be a whole number, we round up to the nearest integer to ensure the desired level of confidence and margin of error. Therefore, each group must have a sample size of at least 348 students.