Statistics 220

Question

Statistics 220

 

A group of realtors estimates that 23% of all homes purchased last year were considered investment properties.  If a sample of 800 homes sold last year is obtained, what is the probability that at most 200 homes are going to used as investment property?

 

0,4066, or 0.5935. or 0.9099, or 0.0901?

 

Thanks,

Dolly

Answer 1

Since a sample was obtained, this problem is working with the sampling distribution of the sample proportion (p-hat).  We will need to use this distribution to find the probability.

 

A sampling distribution of the sample proportion has a mean equal to the population proportion (p).  The standard deviation of the distribution is equal to sqrt ((p(1-p))/n).  The sampling distribution in this problem will be normal because the sample size is large.

 

For this problem, the population proportion (p) is 0.23 (23%/100 = 0.23).  Therefore, the mean of the sampling distribution is 0.23.  The sample size is 800, so the standard deviation is sqrt ((0.23(1-0.23))/800) = sqrt ((0.1771)/800) = 0.0149

 

The sample proportion (p-hat) is 200 out of 800, or 200/800 = 0.25

 

Next, calculate the z-score for the sample proportion 0.25 using the sampling distribution information.

    z = (p-hat - mean) / std dev = (0.25 - 0.23) / 0.0149 = 1.34

 

The last part is to find the probability in a z table for the normal distribution.  The z-score is 1.34.  The probability that "at most 200 homes are used" is the same as the probability that less than 200 homes are used.  This probability is the area to the left of 1.34.  The area to the left of z = 1.34 is 0.9099.

 

The answer is 0.9099.