In a study of children with a particular disorder, parents were asked to rate their child on a variety of items related to how well their child performs different tasks. One item was "Has difficulty organizing work," rated on a five-point scale of 0 to 4 with 0 corresponding to "not at all" and 4 corresponding to "very much." The mean rating for 275 boys with the disorder was reported as 2.31 with a standard deviation of 1.11. (Round your answers to four decimal places.) Find 90%, 95%, and 99% confidence intervals.

Question

In a study of children with a particular disorder, parents were asked to rate their child on a variety of items related to how well their child performs different tasks. One item was "Has difficulty organizing work," rated on a five-point scale of 0 to 4 with 0 corresponding to "not at all" and 4 corresponding to "very much." The mean rating for 275 boys with the disorder was reported as 2.31 with a standard deviation of 1.11. (Round your answers to four decimal places.) Find 90%, 95%, and 99% confidence intervals.

Answer 1

To find the confidence intervals for the mean rating, we can use the formula:

Confidence Interval = Mean ± (Z * (Standard Deviation / √n))

Where:
- Mean is the average rating for the sample
- Z is the z-score corresponding to the desired confidence level
- Standard Deviation is the standard deviation of the sample
- n is the sample size

Let's calculate the confidence intervals for the given data:

1. For a 90% confidence interval:

First, we need to find the z-score corresponding to a 90% confidence level. This can be looked up in the z-table or calculated using statistical software. For a 90% confidence level, the z-score is approximately 1.645.

Confidence Interval = 2.31 ± (1.645 * (1.11 / √275))

Calculating the values:
Confidence Interval = 2.31 ± (1.645 * 0.0666)

Lower bound = 2.31 - 0.1092
Upper bound = 2.31 + 0.1092

Result: The 90% confidence interval is (2.2008, 2.4192).

2. For a 95% confidence interval:

For a 95% confidence level, the z-score is approximately 1.96.

Confidence Interval = 2.31 ± (1.96 * (1.11 / √275))

Calculating the values:
Confidence Interval = 2.31 ± (1.96 * 0.0666)

Lower bound = 2.31 - 0.1305
Upper bound = 2.31 + 0.1305

Result: The 95% confidence interval is (2.1795, 2.4405).

3. For a 99% confidence interval:

For a 99% confidence level, the z-score is approximately 2.576.

Confidence Interval = 2.31 ± (2.576 * (1.11 / √275))

Calculating the values:
Confidence Interval = 2.31 ± (2.576 * 0.0666)

Lower bound = 2.31 - 0.1717
Upper bound = 2.31 + 0.1717

Result: The 99% confidence interval is (2.1383, 2.4817).

So, the confidence intervals for the mean rating are:
- 90% confidence interval: (2.2008, 2.4192)
- 95% confidence interval: (2.1795, 2.4405)
- 99% confidence interval: (2.1383, 2.4817)

Answer 2

Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability (±.05, ±.025, ±.005) to get the Z scores.

% = mean ± Z SEm

SEm = SD/√n