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 278 boys with the disorder was reported as 2.28 with a standard deviation of 1.14. (Round your answers to four decimal places.)
Compute the 90% confidence interval.
( 2.167 , 2.393) This is right
Compute the 95% confidence interval.
( 2.1459, 2.4141 ) This is wrong
Compute the 99% confidence interval.
( 2.1035, 2.4565 ) This is wrong
It depends if you are looking at a distribution of scores or a distribution of means.
Since the scores cannot vary more than 2 SD in either direction, I assume that you are looking for means.
90% = mean ± 2.04 SEm
SEm = SD/√n
95% = mean ± 1.96 SEm
99% = mean ± 2.575 SEm
I'll let you do the calculations.
To compute a confidence interval, we need to use the formula:
Confidence Interval = Mean ± (Critical Value) × (Standard Deviation / √Sample Size)
For a confidence level of 90%, the critical value can be found using a standard normal distribution table. The critical value for a 90% confidence level is 1.645.
Substituting the given values into the formula:
Mean = 2.28
Standard Deviation = 1.14
Sample Size = 278
Critical Value = 1.645
Confidence Interval = 2.28 ± 1.645 × (1.14 / √278)
Calculating the confidence interval:
Confidence Interval = 2.28 ± 1.645 × (1.14 / √278)
Confidence Interval ≈ 2.28 ± 0.1061
Confidence Interval ≈ (2.1739, 2.3861)
Therefore, the correct 90% confidence interval is (2.1739, 2.3861).
For the 95% confidence interval and 99% confidence interval, we need to use different critical values.
For a 95% confidence level, the critical value is 1.96.
Confidence Interval = 2.28 ± 1.96 × (1.14 / √278)
Confidence Interval ≈ 2.28 ± 0.1188
Confidence Interval ≈ (2.1612, 2.3988)
Therefore, the correct 95% confidence interval is (2.1612, 2.3988).
For a 99% confidence level, the critical value is 2.576.
Confidence Interval = 2.28 ± 2.576 × (1.14 / √278)
Confidence Interval ≈ 2.28 ± 0.1537
Confidence Interval ≈ (2.1263, 2.4337)
Therefore, the correct 99% confidence interval is (2.1263, 2.4337).
In conclusion, the correct confidence intervals are:
90% confidence interval: (2.1739, 2.3861)
95% confidence interval: (2.1612, 2.3988)
99% confidence interval: (2.1263, 2.4337)
To compute the confidence intervals for the given study, we can use the formula:
Confidence Interval = Mean ± (Critical Value × Standard Error)
The critical value corresponds to the level of confidence desired, which determines the width of the interval. To find the critical value, we can refer to the z-table.
For a 90% confidence interval, the critical value is 1.645.
For a 95% confidence interval, the critical value is 1.96.
For a 99% confidence interval, the critical value is 2.576.
The standard error can be calculated using the formula:
Standard Error = (Standard Deviation) / sqrt(n)
where n is the sample size.
Now, using the given information:
Mean = 2.28
Standard Deviation = 1.14
Sample Size (n) = 278
Computing the standard error:
Standard Error = 1.14 / sqrt(278) = 0.0685 (approx.)
Now let's calculate the confidence intervals.
For a 90% confidence interval:
Confidence Interval = 2.28 ± (1.645 × 0.0685) ≈ (2.167, 2.393)
For a 95% confidence interval:
Confidence Interval = 2.28 ± (1.96 × 0.0685) ≈ (2.1459, 2.4141)
For a 99% confidence interval:
Confidence Interval = 2.28 ± (2.576 × 0.0685) ≈ (2.1035, 2.4565)
Therefore, the correct answers are:
- 90% confidence interval: (2.167, 2.393)
- 95% confidence interval: (2.1459, 2.4141)
- 99% confidence interval: (2.1035, 2.4565)