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.

Question

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.

Answer 1

To compute a 90% confidence interval for the population parameters, we can use the formula:

Confidence interval = sample mean ± (critical value * standard error)

1. Let's start by calculating the critical value using the standard normal distribution. Since we want a 90% confidence interval, we need to find the z-value corresponding to a confidence level of 0.90. This value can be looked up in a standard normal distribution table or calculated using statistical software. The critical value for a 90% confidence interval is approximately 1.645.

2. Next, we need to calculate the standard error of the mean. The standard error (SE) represents the average deviation of sample means from the population mean. It is calculated by dividing the standard deviation of the sample by the square root of the sample size.

SE = standard deviation / √(sample size)

3. We are given the sum of squares (SS) for the sample, but we need to calculate the sample variance (s²) in order to obtain the standard deviation. The sample variance can be calculated by dividing SS by the sample size minus 1 (n - 1).

s² = SS / (n - 1)

4. Finally, we can substitute the values we have obtained into the confidence interval formula:

Confidence interval = sample mean ± (critical value * standard error)

Confidence interval = 43 ± (1.645 * (s / √n))

Let's gather the values we need to calculate the confidence interval:

Sample mean (x̄) = 43
Sample size (n) = 25
SS = 2400
Critical value (z) = 1.645

First, we calculate the sample variance:

s² = SS / (n - 1)
= 2400 / (25 - 1)
= 2400 / 24
= 100

Next, we calculate the standard deviation:

s = √s²
= √100
= 10

Now we can calculate the standard error:

SE = s / √n
= 10 / √25
= 10 / 5
= 2

Finally, we calculate the confidence interval:

Confidence interval = 43 ± (1.645 * (s / √n))

Confidence interval ≈ 43 ± (1.645 * 2)

Confidence interval ≈ 43 ± 3.29

Therefore, using a 90% confidence interval, we can estimate that the population parameters for the test scores lie between 39.71 and 46.29.