Question

Health insurers are beginning to offer telemedicine services online that replace the common office visit. A company provides a video service that allows subscribers to connect with a physician online and receive prescribed treatments. The company claims that users of its 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 the company.

92 34 40
105 83 55
56 49 40
76 48 96
93 74 73
78 93 100
53 82

Required:
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.

Answer 1

`\text {CI} = (60.54, \: 81.46)\\\\`

Therefore, we are 95% confident that actual mean savings for a televisit to the doctor is within the interval of ($60.54 to $81.46)

Explanation:

Let us find out the mean savings for a televisit to the doctor from the given data.

Using Excel,

=AVERAGE(number1, number2,....)

The mean is found to be

`\bar{x} = \$71`

Let us find out the standard deviation of savings for a televisit to the doctor from the given data.

Using Excel,

=STDEV(number1, number2,....)

The standard deviation is found to be

`s = \$ 22.35`

The confidence interval is given by

`\text {confidence interval} = \bar{x} \pm MoE\\\\`

Where the margin of error is given by

`$ MoE = t_(\alpha/2)((s)/(√(n) ) ) $ \\\\`

Where n is the sample of 20 online doctor visits, s is the sample standard deviation and
`t_(\alpha/2)` is the t-score corresponding to a 95% confidence level.

The t-score is given by is

Significance level = α = 1 - 0.95 = 0.05/2 = 0.025

Degree of freedom = n - 1 = 20 - 1 = 19

From the t-table at α = 0.025 and DoF = 19

t-score = 2.093

So, the margin of error is

`MoE = t_(\alpha/2)((s)/(√(n) ) ) \\\\MoE = 2.093\cdot (22.35)/(√(20) ) \\\\MoE = 2.093\cdot 4.997\\\\MoE = 10.46\\\\`

So the required 95% confidence interval is

`\text {CI} = \bar{x} \pm MoE\\\\\text {CI} = 71 \pm 10.46\\\\\text {CI} = 71 - 10.46, \: 71 + 10.46\\\\\text {CI} = (60.54, \: 81.46)\\\\`

Therefore, we are 95% confident that actual mean savings for a televisit to the doctor is within the interval of ($60.54 to $81.46)