Explanation:
Step 1
The problem is asking for a 95% confidence interval for the average difference between the reading and writing scores of all students. The mean difference between the scores, the sample size, and the standard deviation are provided.
Step 2
The formula for a confidence interval is given by:
( CI = X ± (Z * \frac{σ}{\sqrt{n}}) )
Where:
( X ) is the sample mean,
( σ ) is the population standard deviation,
( n ) is the sample size, and
( Z ) is the Z score for the desired confidence level.
Step 3
For a 95% confidence level, the Z score is approximately 1.96.
Step 4
Substitute the given values into the formula:
( X = -0.545 )
( Z = 1.96 )
( n = 200 )
( σ = 8.887 )
Step 5
Calculate the lower and upper bounds of the confidence interval.
Answer:
(a) lower bound = -2.083
(b) upper bound = 0.940
First, we need to calculate the standard error (SE) of the mean difference. The formula for SE is given by:
[SE = \frac{s}{\sqrt{n}}]
where (s) is the standard deviation and (n) is the sample size.
Step 2:
Next, we calculate the confidence interval using the formula:
[CI = \mu \pm Z_{\alpha/2} * SE]
where (\mu) is the mean, (Z{\alpha/2}) is the z-score corresponding to the desired level of confidence (for a 95% confidence interval, (Z{\alpha/2} = 1.96)), and SE is the standard error calculated in step 1.
Calculations:
Step 1:
Given that the standard deviation (s = 8.887) and the sample size (n = 200), we can calculate the standard error as follows:
[SE = \frac{8.887}{\sqrt{200}} = 0.6287]
Step 2:
Given that the mean (\mu = -0.545), we can calculate the confidence interval as follows:
[CI = -0.545 \pm 1.96 * 0.6287]
[CI = -0.545 \pm 1.2323]
This gives us a lower bound of (-1.7773) and an upper bound of (0.6873).
Answer:
(a) Lower bound: -1.78 points
Upper bound: 0.69 points
(b) We can be 95% confident that the average difference between reading and writing scores of all students is contained within our confidence interval.
(c) No, since 0 is contained in our confidence interval, we cannot conclude that there is a real difference in the average scores.
