A simple random sample of students is selected, and the students are asked how much time they spent
preparing for a test. The times (in hours) are as follows: 1.3 7.2 4.2 12.5 6.6 2.5 5.5 Based on these results, a confidence interval for the population mean is found to be µ = 5.7 ± 4.4. Find the degree of confidence.
Random sample of 17 HMO was selected. For each HMO the copy's for a doctors visit was recorded as 9,7,10,9,8,12,6,5,5,11,7,6,7,10,12,8,6 with s 95% confidence interval for the mean co payment
To find the degree of confidence, we need to look at the format of the confidence interval. The confidence interval is given as µ = 5.7 ± 4.4.
The format of a confidence interval is usually expressed as:
µ ± margin of error
In this case, the margin of error is 4.4, which means that the interval is 4.4 units above and below the sample mean (5.7).
To determine the degree of confidence, we need to find the corresponding critical value (also known as the Z-score) for this margin of error. This critical value indicates the number of standard deviations that encompass the specified confidence level.
To find the Z-score, we can use a standard normal distribution table or a calculator. The Z-score depends on the desired confidence level.
Let's assume the desired confidence level is denoted as C. The formula to find the Z-score is as follows:
Z = (1 - C) / 2
Since the confidence interval is given in terms of ± (plus-minus), we divide the confidence level by 2 before subtracting it from 1.
For example, if the desired confidence level is 0.95, then:
Z = (1 - 0.95) / 2 = 0.025
Using a standard normal distribution table or calculator, we can find that for a Z-score of 0.025, the corresponding critical value is approximately 1.96 (rounded to two decimal places).
Therefore, the degree of confidence for this confidence interval is approximately 95%.
To determine the degree of confidence, we need to look at the confidence interval provided, which is µ = 5.7 ± 4.4.
A confidence interval provides a range of values within which we are confident that the population mean falls. The interval is usually represented in the form of x̄ ± E, where x̄ is the sample mean and E is the margin of error.
In this case, the confidence interval is µ = 5.7 ± 4.4. The margin of error is 4.4, which means that the population mean could be as high as 5.7 + 4.4 = 10.1 or as low as 5.7 - 4.4 = 1.3.
Now, to determine the degree of confidence, we need to find the critical value associated with the given margin of error. This critical value depends on the desired level of confidence.
Common levels of confidence used are 90%, 95%, and 99%. These percentages correspond to the area under the standard normal distribution curve. For example, a 95% confidence level implies that the area under the curve, which represents the range of possible values, is 95%.
To find the critical value, we need to use a Z-table or a calculator that provides the values for the standard normal distribution. The critical value is the number of standard deviations needed to capture the desired level of confidence.
In this case, since we are not given the critical value directly, we can reverse-engineer it by using the margin of error and the standard deviation.
The formula to calculate the margin of error is E = Z * (σ/√n), where Z is the critical value, σ is the population standard deviation (which we don't have in this case), and n is the sample size.
From the given information, we don't have the population standard deviation, but we have a sample. We can use the sample standard deviation as an estimate for the population standard deviation.
Given that our sample only consists of 7 observations, we can use a t-distribution instead of a normal distribution to calculate the critical value. This is because for small sample sizes, the t-distribution is more appropriate.
We need to estimate the degrees of freedom to determine the specific t-value to use. For a simple random sample, the degrees of freedom are equal to n - 1, where n is the sample size.
In this case, the degrees of freedom are 7 - 1 = 6.
Using this information, we can now determine the critical t-value associated with the sample size and the desired confidence level.
From the provided information, a 95% confidence level is used, so we need to find the t-value that will enclose 95% of the area under the t-distribution curve with 6 degrees of freedom.
Using a t-table or a calculator, we find that the critical t-value is approximately 2.447.
Since we have determined the critical t-value, we can now use it to find the degree of confidence associated with the confidence interval.
The degree of confidence can be calculated using the formula Degree of Confidence = (1 - α) * 100%, where α is the significance level. In this case, α = (1 - confidence level) = (1 - 0.95) = 0.05.
Therefore, the degree of confidence is approximately 1 - 0.05 = 0.95, or 95%.