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 (Bloomberg Businessweek (March 4 – March 9, 2014). 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.
92 34 40 105 83 55 56 49 40 76 48 96 93 74 73 78 93 100 53 82
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
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, we can use the sample data provided and assume a normal distribution.
First, we calculate the mean (x̄) and standard deviation (s) of the sample:
Mean (x̄) = (92 + 34 + 40 + 105 + 83 + 55 + 56 + 49 + 40 + 76 + 48 + 96 + 93 + 74 + 73 + 78 + 93 + 100 + 53 + 82) / 20
= 1367 / 20
= 68.35
Standard Deviation (s) = sqrt(((92-68.35)^2 + (34-68.35)^2 + (40-68.35)^2 + (105-68.35)^2 + (83-68.35)^2 + (55-68.35)^2 + (56-68.35)^2 + (49-68.35)^2 + (40-68.35)^2 + (76-68.35)^2 + (48-68.35)^2 + (96-68.35)^2 + (93-68.35)^2 + (74-68.35)^2 + (73-68.35)^2 + (78-68.35)^2 + (93-68.35)^2 + (100-68.35)^2 + (53-68.35)^2 + (82-68.35)^2) / (20-1))
= sqrt(21778.45/19)
= sqrt(1141.5)
= 33.80
Next, we calculate the standard error (SE) of the mean:
SE = s / sqrt(n)
= 33.80 / sqrt(20)
≈ 7.56
Using the t-distribution with (n-1) degrees of freedom and a 95% confidence level, we can find the critical value. Since the sample size is 20, the degrees of freedom is (20-1) = 19.
The critical value for a 95% confidence level with 19 degrees of freedom is approximately 2.093.
To construct the confidence interval, we use the formula:
CI = x̄ ± (critical value * SE)
CI = 68.35 ± (2.093 * 7.56)
= 68.35 ± 15.81
Therefore, the 95% confidence interval for the mean savings for a televisit to the doctor is approximately $52.54 to $84.16 per visit.
To construct a 95% confidence interval for the mean savings for a televisit to the doctor, we can use the sample data provided.
Step 1: Find the sample mean (x̄) and sample standard deviation (s).
The sample mean can be calculated by adding up all the values and dividing by the sample size. In this case, the sample mean is:
x̄ = (92 + 34 + 40 + 105 + 83 + 55 + 56 + 49 + 40 + 76 + 48 + 96 + 93 + 74 + 73 + 78 + 93 + 100 + 53 + 82) / 20 = 72.6
The sample standard deviation can be calculated as follows:
s = √[Σ(x - x̄)^2 / (n - 1)], where Σ represents the sum of the values.
s = √[(92 - 72.6)^2 + (34 - 72.6)^2 + ... + (82 - 72.6)^2] / (20 - 1) = 23.93
Step 2: Determine the critical value.
For a 95% confidence interval, we need to find the z-value corresponding to a confidence level of 0.95. By using a standard normal distribution table or a calculator, we can find that the critical value is approximately 1.96.
Step 3: Calculate the margin of error.
The margin of error can be calculated using the formula:
Margin of error = Critical value * (Standard deviation / √(Sample size))
Margin of error = 1.96 * (23.93 / √20) = 10.64
Step 4: Determine the confidence interval.
The confidence interval can be calculated as follows:
Lower bound = Sample mean - Margin of error
Lower bound = 72.6 - 10.64 = 61.96
Upper bound = Sample mean + Margin of error
Upper bound = 72.6 + 10.64 = 83.24
Therefore, the 95% confidence interval for the mean savings for a televisit to the doctor can be expressed as $61.96 to $83.24 per visit.
Calculate mean and SD of your distribution.
95% = Mean ± 1.96 SD