Question

A real estate investor thinks the real estate market has bottomed out. One of the variables he examined to arrive at this conclusion was the proportion of houses sold at or above the asking price. In 2010, the proportion of houses sold at or above the asking price was 14%. The real estate investor takes a random sample of 40 recently sold houses and finds that nine of them are selling at or above the asking price. Specify the null and alternative hypotheses to determine whether the proportion of houses sold at or above the asking price has increased. At the 10% significance level, can you conclude that the proportion of houses sold at or above the asking price has increased? (Make sure to follow all procedures for hypothesis testing to answer this)

Answer 1

Explanation:

Hello!

Considering the variables of interest,

X: number of houses sold at or above the asking price.

In 2010 the proportion was 14% (You have to consider this previously known information as the value of the population parameter)

A sample of n=40 recently sold houses was taken and 9 of them are selling at or above asking price.

The proportion of these houses, sample proportion, sold at or above the asking price is p'= x/n= 9/40= 0.225

If what the real state investor is true, then the market can only increase, then the proportion of houses sold at or above asking price will be higher than 14%, symbolically: p>0.14

The hypotheses are:

H₀: p ≤ 0.14

H₁:p > 0.14

α: 0.10

`Z_(H_0)= \frac{p'-p}{\sqrt{(p(1-p))/(n) } } = \frac{0.225-0.14}{\sqrt{(0.14*0.86)/(40) } }= 1.05`

Using the critical value approach, the rejection region is one-tailed to the right, meaning that you'll reject the null hypothesis to big values of Z.

The critical vlaue is
`Z_(1-\alpha )= Z_(0.90)= 1.283`

If
`Z_(H_0)` ≥ 1.283 ⇒ Reject the null hypothesis.

If
`Z_(H_0)` < 1.283 ⇒ Do not reject the null hypothesis.

The calculated value is less than the critical value, the decision is to reject the null hypothesis.

At a 10% level, you can conclude that the proportion of houses sold at or above the asking price at most 14%, meaning that the market hasn't recovered.

I hope you have a SUPER day!

Answer 2

Explanation:

We would set up the hypothesis test. This is a test of a single population mean since we are dealing with mean

a) For the null hypothesis,

p = 0.14

For the alternative hypothesis,

p > 0.14

It is a right tailed test

Considering the population proportion, probability of success, p = 0.14

q = probability of failure = 1 - p

q = 1 - 0.14 = 0.86

Considering the sample,

Sample proportion, p = x/n

Where

x = number of success = 9

n = number of samples = 40

P = 9/40 = 0.225

We would determine the test statistic which is the z score

z = (P - p)/√pq/n

z = (0.225 - 0.14)/√(0.14 × 0.86)/40 = 1.55

Since it is a right tailed test, we would determine the p value for the area above z = 1.55 from the normal distribution table.

P value = 1 - 0.9394 = 0.0606

Since the significance level, 0.1 > the p value, 0.0606, then we would reject the null hypothesis.

Therefore, at the 10% significance level, you can conclude that the proportion of houses sold at or above the asking price has increased.