Health insurers are beginning to offer telemedicine services online that replace the common office visit. Wellpoint provides a video service that allows subscribers to connect with a physician online and receive prescribed treatments. Wellpoint claims that users of its LiveHealth Online service saved a significant amount of money on a typical visit. The data shown below ($), for a sample of 20 online doctor visits, are consistent with the savings per visit reported by Wellpoint.
90 34 41
107 84 55
56 49 40
76 50 94
94 72 71
80 90 100
53 81
a.Assuming the population is roughly symmetric, construct a 95% confidence interval for the mean savings for a televisit to the doctor as opposed to an office visit (to 2 decimals).
95% confidence interval: $___ to $___per visit
To construct a 95% confidence interval for the mean savings for a televisit to the doctor as opposed to an office visit, you can follow these steps:
Step 1: Calculate the sample mean.
To find the sample mean, add up all the savings values and divide the sum by the number of observations.
Sample mean = (90 + 34 + 41 + 107 + 84 + 55 + 56 + 49 + 40 + 76 + 50 + 94 + 72 + 71 + 80 + 90 + 100 + 53 + 81) / 20
Step 2: Calculate the sample standard deviation.
The sample standard deviation measures the spread of the data. To find it, subtract the sample mean from each savings value, square each difference, sum them up, and divide by (n-1), where n is the number of observations. Finally, take the square root of the result.
- Subtract the mean: (90 - X̄), (34 - X̄), (41 - X̄), ... ,(81 - X̄)
- Square the differences: (90 - X̄)^2, (34 - X̄)^2, (41 - X̄)^2, ... ,(81 - X̄)^2
- Sum them up: (90 - X̄)^2 + (34 - X̄)^2 + (41 - X̄)^2 + ... + (81 - X̄)^2
- Divide by (n-1), where n is 20 in this case.
- Take the square root of the result.
Step 3: Determine the critical value.
For a 95% confidence interval, you need to find the critical value corresponding to a 2-tailed t-distribution with n-1 degrees of freedom. Since the sample size is 20, the degrees of freedom are 20 - 1 = 19. The critical value can be found using a t-distribution table or statistical software. Let's assume the critical value is t*.
Step 4: Calculate the margin of error.
The margin of error is the product of the critical value and the standard error. The standard error is the sample standard deviation divided by the square root of the sample size.
Standard Error = Sample standard deviation / sqrt(n)
Step 5: Calculate the confidence interval.
The confidence interval is calculated by subtracting the margin of error from the sample mean to get the lower bound and adding it to the sample mean to get the upper bound.
Lower bound = Sample mean - (margin of error)
Upper bound = Sample mean + (margin of error)
Finally, substituting the values into the formula will give you the 95% confidence interval for the mean savings per visit.
95% confidence interval: $___ to $___ per visit