Given the following information, determine the 68.3 percent, 95.5 percent, and 99.7 percent confidence intervals.
overbar above X equals 4.33 comma SE sub m equals 3
I would appreciate it extremely if someone would help me figure this out. I honestly have no idea how to do this. Help please I am lost!
I noticed someone else posted this exact question and the answer was to look at the back of a table, however I have an electronic version which doesn't have it. Help please
% = mean ± Z(SEm)
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability (%/2) from the mean and its Z score. Substitute values into above equation and calculate.
If you don't have that table in the back of your text, try this:
https://www.google.com/search?client=safari&rls=en&q=z+scores+calculator&ie=UTF-8&oe=UTF-8
To determine the confidence intervals, we need to use the information given about the sample mean (X̄) and the standard error (SE).
For a given confidence level, the confidence interval can be calculated using the formula:
CI = X̄ ± (Z * SE)
where Z represents the standard score (also known as the z-score) corresponding to the desired confidence level.
To find the Z-score, we can use a standard normal distribution table or a statistical calculator. Since you don't have a table available, we will use a statistical calculator for this example.
Let's calculate the confidence intervals for the given information:
1. For a 68.3% confidence interval:
First, find the z-score corresponding to a 68.3% confidence level. As the distribution is symmetrical, we can use half of the remaining area (100% - 68.3% = 31.7%) on one side of the curve. This gives a tail area of 15.85%.
Using a statistical calculator, the z-score for a tail area of 15.85% is approximately 1.
Now we can calculate the confidence interval:
CI = X̄ ± (Z * SE)
= 4.33 ± (1 * 3)
= 4.33 ± 3
= (1.33, 7.33)
Therefore, the 68.3% confidence interval is (1.33, 7.33).
2. For a 95.5% confidence interval:
Similar to the previous step, find the z-score corresponding to a 95.5% confidence level. As before, we consider half of the remaining area (100% - 95.5% = 4.5%) on one side of the curve, which gives a tail area of 2.25%.
Using a statistical calculator, the z-score for a tail area of 2.25% is approximately 2.
Now we can calculate the confidence interval:
CI = X̄ ± (Z * SE)
= 4.33 ± (2 * 3)
= 4.33 ± 6
= (-1.67, 10.33)
Therefore, the 95.5% confidence interval is (-1.67, 10.33).
3. For a 99.7% confidence interval:
Again, find the z-score corresponding to a 99.7% confidence level. The remaining area on one side of the curve is 0.3%, which means a tail area of 0.15%.
Using a statistical calculator, the z-score for a tail area of 0.15% is approximately 3.
Now we can calculate the confidence interval:
CI = X̄ ± (Z * SE)
= 4.33 ± (3 * 3)
= 4.33 ± 9
= (-4.67, 13.33)
Therefore, the 99.7% confidence interval is (-4.67, 13.33).
To summarize:
- The 68.3% confidence interval is (1.33, 7.33).
- The 95.5% confidence interval is (-1.67, 10.33).
- The 99.7% confidence interval is (-4.67, 13.33).
To determine confidence intervals, we need to know the sample mean (X̄), the standard error of the mean (SEm), and the level of confidence (often expressed as a percentage). In this case, you are given X̄ = 4.33 and SEm = 3.
Confidence intervals are usually determined using the standard normal distribution (z-distribution) or the t-distribution. Since the sample size is not provided, we'll assume it's large enough (more than 30) to use the z-distribution.
The formula for the confidence interval is:
CI = X̄ ± (z * SEm)
First, we need to find the z-score corresponding to each level of confidence. The z-scores for commonly used levels of confidence are:
- 68.3 percent confidence interval: z = 1.00 (approximately)
- 95.5 percent confidence interval: z = 2.00 (approximately)
- 99.7 percent confidence interval: z = 3.00 (approximately)
Now, let's calculate the confidence intervals using the formula above:
For the 68.3 percent confidence interval:
CI = 4.33 ± (1.00 * 3)
= 4.33 ± 3
= (1.33, 7.33)
For the 95.5 percent confidence interval:
CI = 4.33 ± (2.00 * 3)
= 4.33 ± 6
= (-1.67, 10.33)
For the 99.7 percent confidence interval:
CI = 4.33 ± (3.00 * 3)
= 4.33 ± 9
= (-4.67, 13.33)
Therefore, the confidence intervals are as follows:
- 68.3 percent confidence interval: (1.33, 7.33)
- 95.5 percent confidence interval: (-1.67, 10.33)
- 99.7 percent confidence interval: (-4.67, 13.33)
Please note that these calculations assume a large sample size and the use of the z-distribution. If the sample size is small or if you are using the t-distribution, the process would be slightly different.