Questions: A certain counselor wants to compare mean IQ scores for two different social groups. A random sample of 16 IQ scores from group 1 showed a mean of 89 and a standard deviation of 14, while an independently chosen random sample of 15 IQ scores from group 2 showed a mean of 90 and a standard deviation of 13. Assuming that the populations of IQ scores are normally distributed for each of the groups and that the variances of these populations are equal, construct a 90% confidence interval for the difference μ₁-μ₂ between the mean μ₁ of IQ scores of group 1 and the mean μ₂ of IQ scores of group 2. Then find the lower limit and upper limit of the 90% confidence interval. Carry your intermediate computations to at least three decimal places. Round your responses to at least two decimal places. (If necessary, consult a list of formulas.) Lower limit: Upper limit:

A certain counselor wants to compare mean IQ scores for two different social groups. A random sample of 16 IQ scores from group 1 showed a mean of 89 and a standard deviation of 14, while an independently chosen random sample of 15 IQ scores from group 2 showed a mean of 90 and a standard deviation of 13. Assuming that the populations of IQ scores are normally distributed for each of the groups and that the variances of these populations are equal, construct a 90% confidence interval for the difference μ₁-μ₂ between the mean μ₁ of IQ scores of group 1 and the mean μ₂ of IQ scores of group 2. Then find the lower limit and upper limit of the 90% confidence interval.

Carry your intermediate computations to at least three decimal places. Round your responses to at least two decimal places. (If necessary, consult a list of formulas.)

Lower limit:

Upper limit:

Transcript text: A certain counselor wants to compare mean IQ scores for two different social groups. A random sample of 16 IQ scores from group 1 showed a mean of 89 and a standard deviation of 14, while an independently chosen random sample of 15 IQ scores from group 2 showed a mean of 90 and a standard deviation of 13. Assuming that the populations of IQ scores are normally distributed for each of the groups and that the variances of these populations are equal, construct a $90\%$ confidence interval for the difference $\mu_{1}-\mu_{2}$ between the mean $\mu_{1}$ of IQ scores of group 1 and the mean $\mu_{2}$ of IQ scores of group 2. Then find the lower limit and upper limit of the $90\%$ confidence interval. Carry your intermediate computations to at least three decimal places. Round your responses to at least two decimal places. (If necessary, consult a list of formulas.) Lower limit: Upper limit:

Solution

Solution Steps

Step 1: Given Data

We have two independent groups with the following statistics:

Group 1:
- Sample size $ n_1 = 16 $
- Mean $ \mu_1 = 89 $
- Standard deviation $ \sigma_1 = 14 $
Group 2:
- Sample size $ n_2 = 15 $
- Mean $ \mu_2 = 90 $
- Standard deviation $ \sigma_2 = 13 $

Step 2: Calculate the Mean Difference

The mean difference between the two groups is calculated as: \[ \text{Mean Difference} = \mu_1 - \mu_2 = 89 - 90 = -1 \]

Step 3: Calculate the Pooled Standard Deviation

The pooled standard deviation $ \sigma_p $ is calculated using the formula: \[ \sigma_p = \sqrt{\frac{(n_1 - 1) \sigma_1^2 + (n_2 - 1) \sigma_2^2}{n_1 + n_2 - 2}} \] Substituting the values: \[ \sigma_p = \sqrt{\frac{(16 - 1) \cdot 14^2 + (15 - 1) \cdot 13^2}{16 + 15 - 2}} = 13.5265 \]

Step 4: Calculate the Z-Score for the Confidence Level

For a $ 90\% $ confidence level, the Z-score is: \[ Z = 1.6449 \]

Step 5: Calculate the Margin of Error

The margin of error $ E $ is calculated using the formula: \[ E = Z \cdot \frac{\sigma_p}{\sqrt{n_1 + n_2}} = 1.6449 \cdot \frac{13.5265}{\sqrt{31}} = 3.9961 \]

Step 6: Construct the Confidence Interval

The $ 90\% $ confidence interval for the difference in means is given by: \[ \text{Confidence Interval} = \left( \text{Mean Difference} - E, \text{Mean Difference} + E \right) \] Substituting the values: \[ \text{Confidence Interval} = (-1 - 3.9961, -1 + 3.9961) = (-4.9961, 2.9961) \]

Step 7: Round the Limits

Rounding the limits to two decimal places: