A sample of 300 is taken and the sample mean is computed to be 7.65 and the sample standard deviation is 5. Assume the data is ratio level.
Compute the 95% confidence interval on the mean.
Compute the 3.9% confidence interval on the mean
Which confidence interval is more practical and useful in a business situation? Why?
To compute confidence intervals, we need to use the formula:
Confidence Interval = sample mean ± (critical value) × (sample standard deviation / √sample size)
1. Compute the 95% confidence interval:
First, we need to find the critical value associated with a 95% confidence level. Since the data is ratio level and the sample size is large (n > 30), we can use the Z-distribution and the standard normal table. For a 95% confidence level, the critical value is 1.96.
95% confidence interval = 7.65 ± (1.96) × (5 / √300)
Simplifying the equation:
95% confidence interval = 7.65 ± (1.96) × (5 / √300)
95% confidence interval ≈ (6.14, 9.16)
Therefore, the 95% confidence interval on the mean is approximately (6.14, 9.16).
2. Compute the 3.9% confidence interval:
To compute the 3.9% confidence interval, we need to find the critical value associated with a 3.9% confidence level. Again, we can use the Z-distribution and the standard normal table. The critical value for a 3.9% confidence level is -1.97.
3.9% confidence interval = 7.65 ± (-1.97) × (5 / √300)
Simplifying the equation:
3.9% confidence interval = 7.65 ± (-1.97) × (5 / √300)
3.9% confidence interval ≈ (6.17, 9.13)
Therefore, the 3.9% confidence interval on the mean is approximately (6.17, 9.13).
3. Which confidence interval is more practical and useful in a business situation? Why?
In a business situation, the 95% confidence interval is more practical and useful. A 95% confidence interval is commonly used in statistical analysis as it provides a high level of certainty. It means that if the same experiment is repeated multiple times, 95% of the resulting confidence intervals will contain the true population mean. This level of confidence helps to reduce the risk of making incorrect decisions based on the sample mean.
On the other hand, the 3.9% confidence interval is extremely narrow and less commonly used in practice. Such a narrow interval implies a very high level of certainty, but it also comes at the cost of a significant reduction in the range of potential values. This may lead to unnecessarily strict decision-making criteria and limit flexibility in business decisions.
Therefore, the 95% confidence interval is preferred in most business situations as it strikes a balance between providing a reasonable level of certainty and allowing for a wider range of potential values.