Question

1. Suppose there is a hospital that has the following information about the insurance that their patients have. The probabilities below are taken from a previous study from the early 2000's that found that on average this is the distribution of insurers for the hospital.

Type	Percentage
Other	25%
UnitedHealth Group	5%
Cigna	10%
Blue cross	25%
Medicaid	10%
Medicare	25%

The hospital takes a survey of their current patient’s insurance information. The company believes that the data still follows the old distribution from the 2000 study.

Type	Frequency	Expected
Other	84
UnitedHealth Group	57
Cigna	69
Blue cross	74
Medicaid	60
Medicare	127

The hospital wants to use an alpha of 10% for their study. The hospital believes that the distribution of insurers still follows the distribution the hospital has from the 2000's. They want you to test this hypothesis.

What is the null and alternative hypotheses for this type of study? (2)

What is the expected distribution of the data, you can create a chard in excel and post it in the response section. If not you can include it in your written work section: (2)

What is the distribution that we use to test our hypotheses and what are the degrees of freedom for this distribution? (2)

What is the sample Statistic for this distribution? (2)

Do we reject or fail to reject the null hypotheses? (2)

Answer 1

The hypothesis test:

Null hypothesis, Ho: The distribution of insurers still follows the distribution the hospital from the 2000's.

Alternative hypothesis, Ha: The distribution of insurers does not follows the distribution the hospital from the 2000's.  

Table:

Type	frequency (O)	P(x)	Expected (E)=n*p	(O-E)2/E
Other	84	0.25	117.75	9.673567
United Health Group	57	0.05	23.55	47.511783
Cigna	69	0.10	47.10	10.182803
Blue cross	74	0.25	117.75	16.255308
Medicaid	60	0.10	47.10	3.533121
Medicare	127	0.25	117.75	0.726645
Total	471	1.00		87.883227

Test statistic, χ2 = 87.8832

Critical value of χ2:

df = k-1 = 6-1 = 5

At α = 0.10 significance level.

Critical value of χ2 (0.10,5) = 9.236

Since, test statistic χ2 > critical value of χ2

We reject the null hypothesis, Ho.