Question

According to the document Market Survey of Long-Term Care Costs, published by the MetLife Mature Market Institute, the average cost of a private room in a nursing home was $239 per day in 2011. A random sample of 7 nursing homes yielded the following daily costs, in dollars, for a private room in a nursing home for this year:

265 283 266 366 243 334
213 266 140 276
292

At the 5% level of significance, does the data provide enough evidence to conclude that the mean cost of a private room in a nursing home this year has changed from the 2011 mean cost?

Answer 1

We accept H₀. The information is not enough to reject H₀

Explanation:

In 2011

Mean for the cost of a private room in a nursing home μ = 239 $/day

Sample in 2021

265 283 266 366 243 334 213 266 140 276 292

average = x = 267,64

Standard deviation s = 56,31

We assume a normal distribution. Random sample, sample size and the fact that population standard deviation is unknown drive to develop a t -student two-tail test at 5 % of significance level

n = 11 then degree of freedom df = n - 1 df = 10

Then from t-student table df = 10 and α = 0,05 α/2 = 0,025

we get t(c) = 2,228

Hypothesis Test:

Null Hypothesis H₀ x = μ = 239

Alternative Hypothesis Hₐ x ≠ μ

To calculate t(s)

t(s) = ( x - μ ) / s/ √n

t(s) = ( 267,64 - 239 ) / 56,31/√10

t(s) = 28,64 *3,16 / 56,31

t(s) = 1,61

Comparing t(s) and t(c)

t(s) < t(c)

Then t(s) is in the acceptance region we accept H₀ . We did´t found enough evidence to reject H₀