The crisis in the real estate market caused the listing prices of homes in certa

Question

The crisis in the real estate market caused the listing prices of homes in certain areas to fall from previous years. A real estate office would like to sample 48
new listings randomly to test the hypothesis that the current listing price average is less than $242,000,
the average in the previous year. Assume the standard deviation for the price of homes in this market is $45,000.
Complete parts a through d below. Using =0.10,
calculate the probability of a Type II error occurring if the actual average listing is $230,000.

Explanation / Answer

Given that,
Standard deviation, =45000
Sample Mean, X =230000
Null, H0: =242000
Alternate, H1: !=242000
Level of significance, = 0.1
From Standard normal table, Z /2 =1.6449
Since our test is two-tailed
Reject Ho, if Zo < -1.6449 OR if Zo > 1.6449
Reject Ho if (x-242000)/45000/(n) < -1.6449 OR if (x-242000)/45000/(n) > 1.6449
Reject Ho if x < 242000-74020.5/(n) OR if x > 242000-74020.5/(n)
-----------------------------------------------------------------------------------------------------
Suppose the size of the sample is n = 48 then the critical region
becomes,
Reject Ho if x < 242000-74020.5/(48) OR if x > 242000+74020.5/(48)
Reject Ho if x < 231316.06109986 OR if x > 252683.93890014
Implies, don't reject Ho if 231316.06109986 x 252683.93890014
Suppose the true mean is 230000
Probability of Type II error,
P(Type II error) = P(Don't Reject Ho | H1 is true )
= P(231316.06109986 x 252683.93890014 | 1 = 230000)
= P(231316.06109986-230000/45000/(48) x - / /n 252683.93890014-230000/45000/(48)
= P(0.20262086 Z 3.49242086 )
= P( Z 3.49242086) - P( Z 0.20262086)
= 0.9998 - 0.5803 [ Using Z Table ]
= 0.4195
For n =48 the probability of Type II error is 0.4195

Related Questions

Navigate

• Browse (All)
• Browse T
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.