2. A researcher has developed a new questionnaire for measuring dominance and would like to estimate the population parameters for the test scores. The questionnaire is administered to a sample of n = 25 subjects. This sample has an average score of mean = 43 with SS = 2400. Compute using a 90% confidence interval.
To compute the confidence interval for the population parameters using the sample data, follow these steps:
Step 1: Determine the sample mean (x̄) and sample size (n).
In this case, the sample mean is 43 and the sample size is 25.
Step 2: Calculate the standard error (SE).
The standard error measures the variability of the sample mean. It can be calculated using the formula:
SE = standard deviation / √n
However, since the standard deviation is not given, we need to calculate it using the sum of squares (SS) and the sample size:
SS = Σ(X - mean)^2
SS = 2400 (given)
Standard deviation (σ) = √(SS / (n - 1))
SE = σ / √n
Step 3: Determine the critical value (z-value) for the desired confidence level.
Since the confidence level is 90%, the alpha level (α) is 1 - confidence level = 1 - 0.90 = 0.10.
To find the critical value (z-value), we need to look up the z-value for the desired alpha level in the standard normal distribution table. For a 90% confidence level, the z-value is approximately 1.645.
Step 4: Calculate the margin of error (ME).
The margin of error represents the possible deviation between the sample mean and the population mean. It can be calculated using the formula:
ME = z * SE
Step 5: Calculate the confidence interval.
The confidence interval represents the range within which the population parameter is estimated to fall. It can be calculated using the formula:
Confidence Interval = sample mean ± margin of error
Now, let's calculate the confidence interval:
Step 1: Given data
Sample mean (x̄) = 43
Sample size (n) = 25
Step 2: Calculate the standard error (SE)
Standard deviation (σ) = √(SS / (n - 1))
= √(2400 / (25 - 1))
= √(2400 / 24)
= √100
= 10
SE = σ / √n
= 10 / √25
= 10 / 5
= 2
Step 3: Determine the critical value (z-value)
z-value = 1.645 (for a 90% confidence level)
Step 4: Calculate the margin of error (ME)
ME = z * SE
= 1.645 * 2
= 3.29 (rounded to 2 decimal places)
Step 5: Calculate the confidence interval
Confidence Interval = sample mean ± margin of error
= 43 ± 3.29
= (39.71, 46.29)
Therefore, the 90% confidence interval for the population parameter is (39.71, 46.29). This means that we can be 90% confident that the true population mean falls within this range based on the given sample data.