Question

9) Three years ago, the mean price of a single-family home was $243,706. A real estate broker believes that the mean price has decreased since then.

(a) Which of the following is the hypothesis test to be conducted?

A) H0: u = $243,706; H1: u < $243,706

B) H0: u = $243,706; H1: u does not equal $243,706

C) H0: u = $243,706; H1: u > $243,706

(b) Which of the following is a type I error?

A) The broker rejects the hypothesis that the mean price is $243,706, when it is the true mean cost.

B) The broker fails to reject the hypothesis that the mean price is $243,706, when the true mean price is less than $243,706.

C) The broker rejects the hypothesis that the mean price is $243,706, when the true mean price is less than $243,706.

(c) Which of the following is a type II error?

A) The broker fails to reject the hypothesis that the mean price is $243,706, when it is the true mean cost.

B) The broker rejects the hypothesis that the mean price is $243,706, when it is the true mean cost.

C) The broker fails to reject the hypothesis that the mean price is $243,706, when the true mean price is less than $243,706

Answer 1

Concepts and reason

Statistical hypotheses testing: Hypotheses testing is used to make inferences about the population based on the sample data. The hypotheses test consists of null hypothesis and alternative hypothesis.

Null hypothesis: The null hypothesis states that there is no difference in the test, which is denoted by . Moreover, the sign of null hypothesis is equal$(=) $, greater than or equal and less than or equal.

Alternative hypothesis: The hypothesis that differs from the is called alternative hypothesis. This signifies that there is a significant difference in the test. The sign of alternative hypothesis is less than , greater than , or not equal .

For testing the mean,

If the population standard deviation is known then the Z test is used.

If the population standard deviation is not known then t test is used.

Fundamentals

Type I Error: Reject the null hypothesis when it is true, called a type I error. It is also known as level of significance. The type I error is denoted as .

Type II Error: Failing to reject the null hypothesis when the alternative is true, called a type II error. The type II error is denoted as .

(9.a)

The hypothesis is stated below,

From the given information, the mean price of a single-family home was $243,706.

Null hypothesis:

Alternative hypothesis:

(9.b)

The correct options are obtained as shown below:

The type I error defined as rejecting a null hypothesis when it is true. This indicates that the test results that ‘the sample is statistically significant’ when actually it is statistically significant. Hence, sample is statistically significant when in reality it is statistically not significant is an example of type I error.

Based on the above situation, the broker rejects the hypothesis that the mean price is $243,706, when it is the true mean cost.

(9.c)

The correct options are obtained as shown below:

The type II error defined as do not rejecting a null hypothesis when it is false. This indicates that the test results that ‘the sample is not statistically significant’ when actually it is statistically significant. Hence, sample is not statistically significant when in reality it is statistically significant is an example of type II error.

Based on the above situation, the broker fails to reject the hypothesis that the mean price is $243,706, when the true mean price is less than $243,706.

Ans: Part 9.a

The hypotheses are,

Null hypothesis: .

Alternative hypothesis: .

Part 9.b

The broker rejects the hypothesis that the mean price is $243,706, when it is the true mean cost.

Part 9.c

The broker fails to reject the hypothesis that the mean price is $243,706, when the true mean price is less than $243,706.