given the following information, determine the 68.3 percent, 95.5 percent, and 99.7 percent confidence intervals. Mean = 4.33, standard error of the mean =3
Percent = mean ± ? SEm
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportions/probabilities related to one-half of the desired percentages to find the Z score and its distance from the mean. For each interval, include that value as the question mark (?) and calculate.
To determine the confidence intervals, we need to know the critical values associated with each confidence level. The critical values are typically obtained from a standard normal distribution table or a statistical software. However, since you haven't provided the sample size, we'll assume it is large enough to use the standard normal distribution.
For a large sample size, the critical values for different confidence levels are as follows:
For a 68.3% confidence interval:
- The critical value (Z) for the upper end is 0.9945.
- The critical value (Z) for the lower end is -0.9945.
To calculate the confidence interval, we can use the following formula:
Confidence Interval = Mean ± (Z * Standard Error of the Mean)
For a 68.3% confidence interval:
Upper bound = Mean + (Z * Standard Error of the Mean) = 4.33 + (0.9945 * 3) = 7.32
Lower bound = Mean - (Z * Standard Error of the Mean) = 4.33 - (0.9945 * 3) = 1.34
Therefore, the 68.3% confidence interval is [1.34, 7.32].
For a 95.5% confidence interval:
- The critical value (Z) for the upper end is 1.9600.
- The critical value (Z) for the lower end is -1.9600.
For a 95.5% confidence interval:
Upper bound = Mean + (Z * Standard Error of the Mean) = 4.33 + (1.9600 * 3) = 11.25
Lower bound = Mean - (Z * Standard Error of the Mean) = 4.33 - (1.9600 * 3) = -2.59
Therefore, the 95.5% confidence interval is [-2.59, 11.25].
For a 99.7% confidence interval:
- The critical value (Z) for the upper end is 2.9700.
- The critical value (Z) for the lower end is -2.9700.
For a 99.7% confidence interval:
Upper bound = Mean + (Z * Standard Error of the Mean) = 4.33 + (2.9700 * 3) = 13.28
Lower bound = Mean - (Z * Standard Error of the Mean) = 4.33 - (2.9700 * 3) = -4.62
Therefore, the 99.7% confidence interval is [-4.62, 13.28].