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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?

Respuesta :

Answer:

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

Step-by-step 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

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